The term “$\infty$-category” refers to a joint higher generalization of the notion of groupoid, category, and 2-groupoid, 3-groupoid, … ∞-groupoid.
Generalising how in an ordinary category, one has morphisms going between objects, and in a 2-category, one has both morphisms (or 1-morphisms or 1-cells) between objects and 2-morphisms (or 2-cells) going between 1-morphisms, in an $\infty$-category, there are k-morphisms going between $(k-1)$-morphisms for all $k = 1, 2, \ldots$. (The $0$-morphisms are the objects of the $\infty$-category.)
There are two crucially different uses of the term:
If one speaks strictly only of the joint generalization of category and ∞-groupoid, hence of the notion of internal category in homotopy theory, then the “$\infty$-”-prefix is to be read as in A-∞ algebra, E-∞ algebra, L-∞ algebra and, in fact, A-∞ category: in all these cases it means that the defining structural relations such as associativity of morphisms are taken to hold up to coherent higher homotopy, also called strong homotopy.
This meaning of “$\infty$-category” is also, less ambiguously, called (∞,1)-category (following the pattern of (n,r)-categories). For more on this notion turn to the entry (∞,1)-category.
In a more encompassing view on higher category theory one may take the maximal “weakening” of structures as implicit and speak of just 2-category to mean a bicategory or rather a (∞,2)-category, of just 3-category to mean a tricategory or rather a (∞,3)-category, of just 4-category to mean a tetracategory or rather (∞,4)-category, and so on. With this counting then an “$\infty$-category” is some limiting notion of $(\infty,\infty)$-category. With this meaning one also often speaks of ω-categories.
This is hence a much more encompassing notion of $\infty$-category than that of (∞,1)-category. It is also much harder to formalize. While there is by now a very good (∞,1)-category theory/homotopy theory of (∞,n)-categories for all $n \in \mathbb{N}$, the limiting case where $n \to \infty$ is currently still poorly understood. While there are several existing proposed definitions for what a single ω-category is, in the most general sense, there is no real understanding of the correct morphisms between them, hence of the correct (∞,1)-category of ω-categories. But this may of course change with time.
If all the $j$-morphisms in an $\infty$-category are equivalences in some suitable sense, we call the $\infty$-category an ∞-groupoid. In this case we can think of the $j$-morphisms for $j\ge 1$ as “homotopies” and the $\infty$-groupoid as a model for a homotopy type. By analogy, we can, if we wish, think of an arbitrary $\infty$-category as a combinatorial model for a directed homotopy type.
There are many different definitions realizing the general idea of $\infty$-category. Models for $\infty$-categories usually fall into two classes:
in the geometric definition of higher category an $\infty$-category is a conglomerate of geometric shapes for higher structures with extra properties;
in the algebraic definition of higher category an $\infty$-category is a conglomerate of geometric shapes for higher structures with extra structure;
One of the tasks of higher category theory is to relate and organize all these different models to a coherent general theory.
There are many different definitions of $\infty$-categories, which may differ in particular in the degree to which certain structural identities are required to hold as equations or allowed to hold up to higher morphisms.
(For detailed references see at (∞,1)-category, (∞,n)-category, (n,r)-category and ω-category.)
For a very gentle introduction to notions of higher categories, try The Tale of n-Categories, which begins in “week73” of This Week’s Finds and goes on from there… keep clicking the links.
For a slightly more formal but still pathetically easy introduction, try:
For a free introductory text on $n$-categories that’s full of pictures, try this:
Tom Leinster has written about “comparative $\infty$-categoriology” (to borrow a term):
Tom Leinster, A Survey of Definitions of n-Category (arXiv)
Tom Leinster, Higher Operads, Higher Categories (arXiv)
Emily Riehl gave a minicourse on infinity categories at the Young Topologists’ Meeting 2015.
A syntactic definition is given by Eric Finster in terms of opetopic type theory, see there for details