Hard to say with so little info. Can you add some pictures to this post, perhaps tell us when you started them and what kind of soil and light you are giving them? Insufficient light is one of the causes of leggy seedlings but there may be other things going on. Peppers are not like tomato plants in that you don't plant them deep, so, yes, spindly plants can take extra time and effort to harden off and grow once they go outside.

yes, all good question. It is usually one of two things, a problem with the lighting or over watering. Both easily corrected if done soon enough. But if they get too leggy, you run the risk of them toppling over. Too much moisture can cause damping off fungus and that is a bit tougher to deal with as well.

Post some photos and answer the questions posed by aniajs and it will be a lot easier to pin down your exact problem and how best to solve it.

FWIW Peppers at the young stages can be buried deeper like tomato's I have experience with this personally, as long the stem is not wooded yet. Please post pics and describe the growing enviorment as stated above, that will help.

John from youtube made a video a while back about planting peppers deeper and while it didn't kill the plants, the stem didn't root at all.
https://www.youtube.com/watch?v=7_pimdio-i8

he tends to talk a lot, so skip to 1:45 to see the part about planting peppers deeper. i would suggest if planting peppers deeper would be to use rooting hormone on the part of the buried stem to try to get the plant to root on the buried section. peppers will root, but no where near as easy as tomatoes.

OP, what kind of lights? CFLs should be about 3-4 inches away from the plants. either way it shouldn't be a huge problem except you'll most likely be needing to stake them or use a tomato cage.

Stevie what he was showing refers to a larger older plant. When they are still soft stemmed they will root if covered in moist potting mix. Many of my sweet peppers if the humidity was too high in the closet started growing roots out of the stem above the soil line. I planted them deeper into a deeper pot. The roots grew.

If the stalk is already hard like woody then no roots will grow out of it.

Those are nothing more than root starts. Had that part of the stem been under ground, it would have put roots out at those spots.

I always thought it was stem/root primordia. Like this tomato. I will repot this guy deeper to test.

I had the roots do that a few years ago

http://forums.gardenweb.com/discussions/2087635/roots-and-root-nodules