A "dimension" isn't anything all that special. A "dimension" in math/physics-speak is not the same "dimension" that appears in scifi. It's not some alternate world where people wear goatees. A "dimension" is basically a free variable.
Consider phase space:
In classical mechanics, if you have some object, this object has certain properties. This object has a position, denoted by three numbers, and a moment, denoted by three more numbers. In total, the object can now be described by six numbers (three for position, three for momentum). These six numbers can be graphed in a six-"dimensional" graph, called phase-space. The object having a certain position and momentum will occupy one point in phase-space. As it moves in position-space, and as it's momentum changes, the object also moves in phase-space.
There. I've just created a six-dimensional space, and I did it without having to resort to using general relativity or string theory.
I believe that the other "dimensions" in string theory are also spatial dimensions, but I'm not really sure.
EDIT
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