Dimension (from Latin “measured out”) is, in essence, the number of degrees of freedom available for movement in a space. (In common usage, the dimensions of an object are the measurements that define its shape and size. That usage is related to, but different from, what this article is about.)
For example, the space in which we live appears to be 3-dimensional. We can move up, north or west, and movement in any other direction can be expressed in terms of just these three. Moving down is the same as moving up a negative amount. Moving northwest is merely a combination of moving north and moving west.
Some theories predict that the space we live in has in fact many more dimensions (frequently 10, 11 or 26) but that the universe measured along these additional dimensions is subatomic in size. See string theory.
Time is frequently referred to as the “fourth dimension”; time is not the fourth dimension of space, but rather of spacetime. This does not have a Euclidean geometry, so temporal directions are not entirely equivalent to spatial dimensions. A tesseract is an example of a four-dimensional object.