My understanding is that some plants will reject pollen that is like their own. Also some plants (eg. holly and papaya) have male and female plants. One plant has male flowers, the other has female flowers. So they can't self-pollinate.
In the case of peppers, they do cross-pollinate with other varieties, as long as they are cultivars of the same species (most peppers are C.annuum). For example two years ago I grew some sweet paprika and one of the plants produced some fruit which were hotter - they had cross pollinated with some anaheims in the next bed. So the cross-pollination influences the fruit that is produced as well as the plant produced by the seed (the paprika actually seeded itself and the plant that came up was a moderately hot paprika).
I usually grow about half a dozen varieties each year, and do not take any precautions against cross-pollinating other than keeping each variety together - and this is mainly for my own memory. I wouldn't describe it as a big problem - just don't be surprised when you get the odd cherry-shaped bell pepper, a cherry red "chocolate" pepper, hot paprikas, etc.!
re. @bstpierre's comments, I would only note that natural hybrids are much more common with plants than animals. Yes they usually have to be between species in the same genus, but plants like Sarracenia (trumpet pitchers), and Opuntia (prickly pear) form natural viable hybrids (within their own genus) in the wild. The only larger animal I'm aware of that naturally produces viable hybrids would be ducks - which are quite randy creatures anyway! (mules are domestic, but they're also sterile).