What kind of bonds involve 2 atoms sharing 3 electrons

can atoms share an odd number of electrons?

There are two explanations for this. The first has something to do with the quantum mechanical model and anti-bonding orbitals; I'm afraid an explanation for this will raise more questions than it answers, so let's move on to something that I have a chance of actually explaining correctly.

What you might be thinking of is resonance. Through this process, electrons can "flow" through a molecule and allow it to remain stable will not quite maintaining a full bond. There are so many intricacies to this process that it would be too difficult to explain here, but this is the simple version: a compound with alternating (we say, "conjugated") double-bonds can in certain circumstances "share" the electrons in these bonds with surrounding atoms in the same compound. 1,3 Butadiene is a good example of this; I can't exactly represent what it looks like here, you'll just have to take my word for it, I guess.

Hope that helps! If you're taking a chemistry course try asking your professor about resonance. I'm sure he or she would be able to explain it in a better manner than I have!

Sure covalent bond...
Say between two Nitrogen atoms, they would share 3 electrons between them, to seem to have a full outer shell.
This would equate to a triple bond- see the below image
http://dkreutz.basd.k12.wi.us/N2.jpg
Hope this helps :)