When people describe our world as three dimensional, they are simplifying.
The dimensionality of our world matters for the sake of describing locations. If you want to tell someone where a ball is, you might give them a number of meters above, to the left and in front of them. When you do this, you are constructing a 3-D coordinate system with them as the origin (point of reference) and with up, left and forward as cardinal directions. However, it is typically implied that you are referring to the ball's position at the current time, or at the time at which the person might reach it. In this sense, we've neglected to mention the time position, but it is still a factor.
Physics treats the universe's dimensionality as it is -- that is, physicists refer to the universe as being 4 dimensional (or more, but that's another topic).
The Special Theory of Relativity describes how we move through the time dimension. We are always moving in the positive direction, but our speed through it is not constant. In fact, if you could see a 4-D arrow created by adding together the contributions of your speed through each spatial dimension and your speed through time, you would notice that the length of the arrow is always constant. The faster you move through space, the slower you move through time, and vice versa.
The General Theory of Relativity expands on this by describing a 4-D space-time that we all inhabit. We move through this 4 dimensional space, but we tend to perceive the 3-D shadow that it casts. This has some important implications. For example, planets don't orbit the sun because of a force that acts like a rope tying them to the star. They are actually moving in perfectly straight lines through 4-D space-time (as no force is acting upon them), but the gravity of the sun is warping the shape of space time until their 3-D shadows appear to move around the sun.
To better visualize this, imagine a space ship rocketing past a planet. It takes a straight path past the planet, but its shadow on the planet appears to follow a semi-circular path centered on the planet.