In physics we refer to 'time' as the fourth dimension. To a moving objects you also need to refer to its change in position with time -- thus the X, Y, and Z position coordinates are the first three dimensions, and t (time) is the fourth.
However-- if you are talking about 'another dimension of space' -- then the best way to think of 4-D is by comparing how our 3-D would affect a 2-D world. If we as three-dimensional beings interacted with a 'flat-lander' (ie a person who lives in a flat piece of paper like reality) then we would do unexplainable things. If I took a 3-D pencil and pushed it into the 2-D flat paper reality, the flat-lander would first see a point as the pencil enters, then a growing circle expand into a hexagon as the pencil pushes through to the shaft. Then I could lay the pencil sideways on the paper and the flat-lander sees it expand into a long pencil shape.
Thus would be the 4th dimension to us-- four dimensional beings could slip in and out of our reality expanding out of no-where into strange 3-D shapes in our reality -- then shifting out as well. We would only see a sliver of the realty they interact with at any given time.