Bicategories
Context
2Category theory
2category theory
Definitions
Transfors between 2categories
Morphisms in 2categories
Structures in 2categories
Limits in 2categories
Structures on 2categories
Higher category theory
higher category theory
Basic concepts
Basic theorems
Applications
Models
Morphisms
Functors
Universal constructions
Extra properties and structure
1categorical presentations
Bicategories
Idea
A bicategory is a particular algebraic notion of weak 2category (in fact, the earliest to be formulated, and still the one in most common use). The idea is that a bicategory is a category weakly enriched over Cat: the homobjects of a bicategory are homcategories, but the associativity and unity laws of enriched categories hold only up to coherent isomorphism.
Definition
A bicategory $B$ consists of
 A collection of objects $x,y,z,\dots$, also called $0$cells;
 For each pair of $0$cells $x,y$, a category $B(x,y)$, whose objects are called morphisms or $1$cells and whose morphisms are called 2morphisms or $2$cells;
 For each $0$cell $x$, a distinguished $1$cell $1_x\in B(x,x)$ called the identity morphism or identity $1$cell at $x$;
 For each triple of $0$cells $x,y,z$, a functor ${\circ}\colon B(y,z)\times B(x,y) \to B(x,z)$ called horizontal composition;
 For each pair of $0$cells $x,y$, natural isomorphisms called unitors: $\left(
\begin{array}{rcl}
f&\mapsto&f \circ 1_x\\
\theta&\mapsto&\theta \circ 1_{1_x}
\end{array}
\right) \cong id_{B(x,y)} \cong \left(
\begin{array}{rcl}
f&\mapsto&1_y\circ f\\
\theta&\mapsto&1_{1_y} \circ \theta
\end{array}
\right):B(x,y)\rightarrow B(x,y)$
 For each quadruple of $0$cells $w,x,y,z$, a natural isomorphism called the associator between the two functors from $B_{y,z} \times B_{x,y} \times B_{w,x}$ to $B_{w,z}$ built out of ${\circ}$; such that
 Such that the pentagon identity is satisfied by the associators.
 And such that the triangle identity is satisfied by the unitors.
If there is exactly one $0$cell, say $*$, then the definition is exactly the same as a monoidal structure on the category $B(*,*)$. This is one of the motivating examples behind the delooping hypothesis and the general notion of ktuply monoidal ncategory.
Details
Here we spell out the above definition in full detail. Compare to the detailed definition of strict $2$category, which is written in the same style but is simpler.
A bicategory $B$ consists of
 a collection $Ob B$ or $Ob_B$ of objects or $0$cells,
 for each object $a$ and object $b$, a collection $B(a,b)$ or $Hom_B(a,b)$ of morphisms or $1$cells $a \to b$, and
 for each object $a$, object $b$, morphism $f\colon a \to b$, and morphism $g\colon a \to b$, a collection $B(f,g)$ or $2Hom_B(f,g)$ of $2$morphisms or $2$cells $f \Rightarrow g$ or $f \Rightarrow g\colon a \to b$,
equipped with
 for each object $a$, an identity $1_a\colon a \to a$ or $\id_a\colon a \to a$,
 for each $a,b,c$, $f\colon a \to b$, and $g\colon b \to c$, a composite $f ; g\colon a \to c$ or $g \circ f\colon a \to c$,
 for each $f\colon a \to b$, an identity or $2$identity $1_f\colon f \Rightarrow f$ or $\Id_f\colon f \to f$,
 for each $f,g,h\colon a \to b$, $\eta\colon f \Rightarrow g$, and $\theta\colon g \Rightarrow h$, a vertical composite $\theta \bullet \eta\colon f \Rightarrow h$,
 for each $a,b,c$, $f\colon a \to b$, $g,h\colon b \to c$, and $\eta\colon g \Rightarrow h$, a left whiskering $\eta \triangleleft f\colon g \circ f \Rightarrow h \circ f$,
 for each $a,b,c$, $f,g\colon a \to b$, $h\colon b \to c$, and $\eta\colon f \Rightarrow g$, a right whiskering $h \triangleright \eta \colon h \circ f \Rightarrow h \circ g$,
 for each $f\colon a \to b$, a left unitor $\lambda_f\colon f \circ \id_a \Rightarrow f$ and an inverse left unitor $\bar{\lambda}_f\colon f \Rightarrow f \circ \id_a$,
 for each $f\colon a \to b$, a right unitor $\rho_f\colon \id_b \circ f \Rightarrow f$, and an inverse right unitor $\bar{\rho}_f\colon f \Rightarrow \id_b \circ f$, and
 for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, an associator $\alpha_{f,g,h}\colon h \circ (g \circ f) \Rightarrow (h \circ g) \circ f$ and an inverse associator $\bar{\alpha}_{f,g,h}\colon (h \circ g) \circ f \Rightarrow h \circ (g \circ f)$,
such that
 for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\eta \bullet \Id_f$ and $\Id_g \bullet \eta$ both equal $\eta$,
 for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b$, the vertical composites $\iota \bullet (\theta \bullet \eta)$ and $(\iota \bullet \theta) \bullet \eta$ are equal,
 for each $a \overset{f}\to b \overset{g}\to c$, the whiskerings $\Id_g \triangleleft f$ and $g \triangleright \Id_f$ both equal $\Id_{g \circ f }$,
 for each $f\colon a \to b$ and $g \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c$, the vertical composite $(\theta \triangleleft f) \bullet (\eta \triangleleft f)$ equals the whiskering $(\theta \bullet \eta) \triangleleft f$,
 for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b$ and $i\colon b \to c$, the vertical composite $(i \triangleright \theta) \bullet (i \triangleright \eta)$ equals the whiskering $i \triangleright (\theta \bullet \eta)$,
 for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\lambda_g \bullet (\eta \triangleleft \id_a)$ and $\eta \bullet \lambda_f$ are equal,
 for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\rho_g \bullet (\id_b \triangleright \eta)$ and $\eta \bullet \rho_f$ are equal,
 for each $a \overset{f}\to b \overset{g}\to c$ and $\eta\colon h \Rightarrow i\colon c \to d$, the vertical composites $\alpha_{f,g,i} \bullet (\eta \triangleleft (g \circ f))$ and $((\eta \triangleleft g) \triangleleft f) \bullet \alpha_{f,g,h}$ are equal,
 for each $f\colon a \to b$, $\eta\colon g \Rightarrow h\colon b \to c$, and $i\colon c \to d$, the vertical composites $\alpha_{f,h,i} \bullet (i \triangleright (\eta \triangleleft f))$ and $((i \triangleright \eta) \triangleleft f) \bullet \alpha_{f,g,i}$ are equal,
 for each $\eta\colon f \Rightarrow g\colon a \to b$ and $b \overset{h}\to c \overset{i}\to d$, the vertical composites $\alpha_{g,h,i} \bullet (i \triangleright (h \triangleright \eta))$ and $((i \circ h) \triangleright \eta) \bullet \alpha_{f,h,i}$ are equal,
 for each $\eta\colon f \Rightarrow g\colon a \to b$ and $\theta\colon h \Rightarrow i\colon b \to c$, the vertical composites $(i \triangleright \eta) \bullet (\theta \triangleleft f)$ and $(\theta \triangleleft g) \bullet (h \triangleright \eta)$ are equal,
 for each $f\colon a \to b$, the vertical composites $\lambda_f \bullet \bar{\lambda}_f\colon f \Rightarrow f$ and $\bar{\lambda}_f \bullet \lambda_f\colon f \circ \id_a \Rightarrow f \circ \id_a$ equal the appropriate identity $2$morphisms,
 for each $f\colon a \to b$, the vertical composites $\rho_f \bullet \bar{\rho}_f\colon f \Rightarrow f$ and $\bar{\rho}_f \bullet \rho_f\colon \id_b \circ f \Rightarrow \id_b \circ f$ equal the appropriate identity $2$morphisms,
 for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, the vertical composites $\alpha_{f,g,h} \bullet \bar{\alpha}_{f,g,h}\colon (h \circ g) \circ f \Rightarrow (h \circ g) \circ f$ and $\bar{\alpha}_{f,g,h} \bullet \alpha_{f,g,h}\colon h \circ (g \circ f) \Rightarrow h \circ (g \circ f)$ equal the appropriate identity $2$morphisms,
 for each $a \overset{f}\to b \overset{g}\to c$, the vertical composite $(\lambda_g \triangleleft f) \bullet \alpha_{f,\id_b,g}$ equals the whiskering $g \triangleright \rho_f$, and
 for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d \overset{i}\to e$, the vertical composites $((\alpha_{g,h,i} \triangleleft f) \bullet \alpha_{f,h \circ g,i}) \bullet (i \triangleright \alpha_{f,g,h})$ and $\alpha_{f,g,i \circ h}\bullet \alpha_{g \circ f,h,i}$ are equal.
It is quite possible that there are errors or omissions in this list, although they should be easy to correct. The point is not that one would want to write out the definition in such elementary terms (although apparently I just did anyway) but rather that one can.
Examples

Any strict 2category is a bicategory in which the unitors and associator are identities. This includes Cat, MonCat, the algebras for any strict 2monad, and so on, at least as classically conceived.

Categories, anafunctors, and natural transformations, which is a more appropriate definition of Cat in the absence of the axiom of choice, form a bicategory that is not a strict 2category. Indeed, without the axiom of choice, the proper notion of bicategory is anabicategory.

Rings, bimodules, and bimodule homomorphisms are the prototype for many similar examples. Notably, we can generalize from rings to enriched categories.

Objects, spans, and morphisms of spans in any category with pullbacks also form a bicategory.

The fundamental 2groupoid? of a space is a bicategory which is not necessarily strict (although it can be made strict fairly easily when the space is Hausdorff by quotienting by thin homotopy, see path groupoid and fundamental infinitygroupoid). When the space is a CWcomplex, there are easier and more computationally amenable equivalent strict 2categories, such as that arising from the fundamental crossed complex.
Coherence theorems
One way to state the coherence theorem for bicategories is that every bicategory is equivalent to a strict $2$category. This “strictification” is not obtained naively by forcing composition to be associative, but (at least in one construction) by freely adding new composites which are strictly associative. Another way to state the coherence theorem is that every formal diagram of the constraints (associators and unitors) commutes.
Note that $n=2$ is the greatest value of $n$ for which every weak $n$category is equivalent to a fully strict one; see semistrict infinitycategory and Graycategory.
The proof of the coherence theorem is basically the same as the proof of the coherence theorem for monoidal categories. An abstract approach can be found in Power‘s paper “A general coherence result.”
The strictification adjunction between bicategories and strict 2categories can be expressed in terms of 3categories; see Campbell.
Terminology
Classically, “2category” meant strict 2category, with “bicategory” used for the weak notion. This led to the more general use of the prefix “2” for strict (that is, strictly Catenriched) notions and “bi” for weak ones. For example, classically a “2adjunction” means a Catenriched adjunction, consisting of two strict 2functors $F,G$ and a strictly Catnatural isomorphism of categories $D(F X, Y)\cong C(X, G Y)$, while a “biadjunction” means the weak version, consisting of two weak 2functors and a pseudo natural equivalence $D(F X, Y)\simeq C(X, G Y)$. Similarly for “2equivalence” and “biequivalence,” and “2limit” and “bilimit.”
We often use “2category” to mean a strict or weak 2category without prejudice, although we do still use “bicategory” to refer to the particular classical algebraic notion of weak 2category. We try to avoid the more general use of “bi” meaning “weak,” however. For one thing, is it confusing; a “biproduct” could mean a weak 2limit, but it could also mean an object which is both a product and a coproduct (which happens quite frequently in additive categories).
Moreover, in most cases the prefix is unnecessary, since once we know we are working in a bicategory, there is usually no point in considering strict notions at all. Fully weak limits are really the only sensible ones to ask for in a bicategory, and likewise for fully weak adjunctions and equivalences. Even in a strict 2category, while we might need to say “strict” sometimes to be clear, we don't need to say “$2$”, since we know that we are not working in a mere category. (Max Kelly pushed this point.)
When we do have a strict 2category, however, other strict notions can be quite technically useful, even if our ultimate interest is in the weak ones. This is somewhat analogous to the use of strict structures to model weak ones in homotopy theory; see here and here for good introductions to this sort of thing.
Discussion about the use of the term “weak enrichment” above is at weak enrichment.
References
 Jean Bénabou, Introduction to Bicategories, Lecture Notes in Mathematics 47, Springer (1967), pp.177. (doi)
See also the references at 2category.
Formalization in homotopy type theory (see also at internal category in homotopy type theory):