That turns out to be one of those really simple, important questions that's really tough to answer. You're asking about forces between electrical charges. I won't try to explain why there has to be a force with positive and negative charges that attract, because I don’t really know. That question gets close to the edge of science where we have to say ’that’s just how things are.’
What I can explain is why we say that the things that attract electrically have charges with opposite signs, positive and negative. If you look at a collection of things, you find they can be grouped into three batches. Any two copies of some object ("likes") fall into the same one of these 3 batches.
1. Ones that don’t have any electrical forces with any others. We call these electrically neutral, and say their electrical charge is zero, for obvious reasons.
2. A batch which all repel each other.
3. Another batch which all repel each other.
All the objects in the second batch attract all the objects in the third batch and vice-versa.
Why do we say that batches 2 and 3 are "opposites"? If you take any object from batch 2, you can get some object from batch 3 which just cancels its electrical charge, meaning that the combination is neutral. When there are two things which add up to zero, that’s just like positive and negative numbers. So we say that every electrical charge is either a positive or a negative. Which batch we call positive and which one we call negative doesn’t matter, so long as we don't keep switching the names around. As it happens, we call electrons negative and atomic nuclei positive. Whether some big object is positive or negative depends on whether it has too few or too many electrons to cancel the charge of its nuclei.
A nice discussion of this can be found in our answer Plastic and Hair: A Static-ey Situation .
All this might sound like a long-winded and pointless description of the obvious. When you think about other forces, though, you see that it’s not so obvious. For gravity, all objects attract. So for gravity likes attract, and there are no opposites. For nuclear forces, the situation is more complicated, but you need more than three categories of objects. So the nuclear charges (so called "color" charges) can’t even be described by regular numbers.
(republished on 07/13/06)