A vector bundle is a vector space which “continuously varies” over a topological space $X$.
A vector bundle over a space $X$ is a bundle over $X$ which is locally isomorphic to a product with a vector space $V$ as fiber. More precisely, the data is an object $\pi: E \to X$ in $Top/X$ equipped with a vector space structure internal to $Top/X$, consisting of maps
(where $E \times_X E$ denotes the fiber product or pullback of $\pi$ along itself) satisfying vector space axioms. This vector space object must satisfy the local triviality condition: there exists an open cover
and an isomorphism from the pullback $U \times_X E$ to the projection $\pi: U \times V \to U$,
as vector space objects in $Top/X$. The projection $U \times V \to U$ itself is called a trivial (vector) bundle over $U$.
Equivalently, each fiber $E_x$ carries a vector space structure, and there exists an open covering $\{U_\alpha\}_{\alpha \in A}$ of $X$ together with local trivializations: bundle isomorphisms $\phi_{\alpha}$ from a trivial bundle $U_{\alpha} \times V$ to the pullback of $\pi$ along $U_\alpha \hookrightarrow X$:
such that $\phi_{\alpha}$ induces a linear map $V \to E_x$ between the fibers.
In terms of the local trivialization data, there are transition functions
where the $g_{\alpha\beta}(x)$ are linear automorphisms of $V$ and satisfy the Čech 1-cocycle conditions:
In the converse direction, given such a collection $g_{\alpha\beta}: U_{\alpha} \cap U_\beta \to GL(V)$ satisfying the 1-cocycle conditions, there is a vector bundle obtained by pasting local trivial bundles together along the $g_{\alpha\beta}$, namely the coequalizer of a pair
in the category of vector space objects in $Top/X$. Here the restriction of $i$ to the coproduct summands is induced by inclusion:
and the restriction of $\mu$ to the coproduct summands is via the action of the transition functions:
Remarks
In most applications, the ground field of scalars is assumed to be $\mathbb{R}$ or $\mathbb{C}$, although sometimes other fields are allowed, as in the study of algebraic vector bundles.
In most cases (as in K-theory), it is implicitly assumed that the vector space $V$ is finite-dimensional.
In the context of differential topology or differential geometry, one also assumes that $\pi$ is smooth and that the local bundle isomorphisms $\phi_{\alpha}$ are diffeomorphisms.
Vector bundles can also be defined via sheaf theory, which permits easy transport to general Grothendieck toposes. Let $Sh(X)$ be the category of (set-valued) sheaves on $X$. The sheaf of continuous local sections of the product projection
forms a local ring object $R$; when interpreted in the internal logic of $Sh(X)$, it is the Dedekind real numbers object. Then, according to a theorem of Richard Swan, in its sheaf-theoretic incarnation a vector bundle is the same thing as a projective $R$-module.
In one class of models for K-theory – generalized (Eilenberg-Steenrod) cohomology theory – cocycles are represented by $\mathbb{Z}_2$-graded vector bundles (pairs of vector bundles, essentially) modulo a certain equivalence relation. In that context it is sometimes useful to consider a certain variant of infinite-dimensional $\mathbb{Z}_2$-graded vector bundles called vectorial bundles.
Much else to be discussed…
vector bundle, holomorphic vector bundle
Glenys Luke, Alexander S. Mishchenko, Vector bundles and their applications, Math. and its Appl. 447, Kluwer 1998. viii+254 pp. MR99m:55019
А. С. Мищенко, Векторные расслоения и их применения (Russian; A. S. Mishchenko, Vector bundles and their applications) Nauka, Moscow, 1984. 208 pp.
Howard Osborn, Vector bundles. Vol. 1. Foundations and Stiefel-Whitney classes, Pure and Appl. Math. 101, Academic Press 1982. xii+371 pp. MR85e:55001
Dale Husemoller, Fibre bundles, McGraw-Hill 1966 (300 p.); Springer GTM 1975 (327 p.), 1994 (353 p.).
An exposition with an eye towards gauge theory is in section 16.1 of
Raoul Bott, Loring Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer 1982. xiv+331 pp.
Discussion with an eye towards K-theory is in
Max Karoubi, K-theory. An introduction, Grundlehren der Mathematischen Wissenschaften 226, Springer 1978. xviii+308 pp.
Allen Hatcher, Vector bundles and K-Theory, (partly finished book) web