A dimension is a measurable extension in space. There are three commonly perceived dimensions: length, breadth, and width. For example, a cube measuring 10 meters in each dimension will have a volume of 1000 cubic meters, i.e. 10 m x 10 m x 10 m = 1000 m3.
The dimension of a geometric shape is the number of independent parameters needed to specify a point on that object.
For example, a line is 1-dimensional, since if you fix an initial point on the line, and choose an orientation for the line (a notion of right and left), then every point on the line may be identified in terms of a single parameter: that point's distance to the right or left from the initial point.
Similarly, the surface of a sphere is 2-dimensional, since one requires 2 independent parameters to specify the location of a point on the surface: the latitude and longitude.
There are other, more elaborate, definitions of dimension, that measure geometric objects like fractal lines, that would have an infinite 1-dimensional length but a zero 2-dimensional area. Fractals have dimensions that are not integer numbers. Fractals don't exist in nature, but some natural objects behave, approximately, like fractals: a broccoli, a costline or the surface of the human brain are examples of natural objects that look like fractals.
Other mathematical definitions
In linear algebra, a real vector space V said to be n-dimensional if V has a basis of size n. Note that this implies that V is isomorphic to n-dimensional Euclidean space. Thus, in agreement with the geometric definition of dimension given above, V is n-dimensional if every point of V is uniquely determined by n real numbers.
In differential topology, a manifold M is said to be n-dimensional if the tangent space to M at every point is an n-dimensional vector space. Note that this implies that M can be covered by local coordinate systems where each coordinate system looks like a region of n-dimensional Euclidean space. In other words, locally, every point on M can be specified using n independent parameters.
Definition in physics
In classical physics, dimension sometimes is takes to mean freedom degrees or the number of independent variables. For example, the study of the displacement of a pendulus is one-dimesional (because there is only one variable, the (oriented) angle that the pendulus forms with the vertical line), however, the three body problem in celestial mechanics is nine-dimensional (because there are three position vectors, each one in three dimensions). Sometimes (like in hamiltonian mechanics) the velocity is also considered to be a variable; so the pendulus would be two-dimensional, and the three-body problem would have eighteen dimensions.
Even before Relativity, the set of all events is 4-dimensional, since in order to specify an event, one needs to know both the position of the event in space (3 parameters), and the time the event took place (an additional 1 parameter). This predicates the idea (believed by most physicists since the time of Albert Einstein) that space and time form a single continuum, and that time is a fourth dimension. However, time in not measured in units of distance such as meters so this system remains incompatible with Newtonian physics. Quantum physicists propose a different theory, namely that there are more than three dimensions in space itself, perhaps as many as eleven. But dimension four and above are all curled up so tightly that they extend no further than the diameter of a subatomic particle, so humans are unable to notice them with the naked eye.
For example, some models in high-energy physics postulate that spacetime has more than 4-dimensions. For example, in the Kaluza-Klein model, a point is specified by a point in 4D-spacetime, plus a point on a circle attached to that point in 4D-spacetime. Though this model is currently out of favor, it illustrates the basic flavor of additional spatial dimensions in the other models; for example, in string theory.