[bio101011]

I am taking organic chemistry for the first time in college so try to bare with me.

I can see that Carbon bonding with Hydrogen is quite common but I am wondering about Oxygen or Nitrogen.

If Nitrogen has already formed three bonds will it constantly have a lone pair or can it be filled with a hydrogen bond?

Oxygen seems to be pretty open in accepting Hydrogen bonds but I could be wrong.

A lot of the elemental bonding properties seem to be situational to whether it's in an acid or base reaction or etc.; (if that's wrong let me know).

[discodermolide]

Both nitrogen and oxygen can accept H-bonds.
Think about amines (R-NH₂) and alcohols (R-OH). These can form H-Bonds.

[AlphaScent]

If you have three hydrogens bonded (3 hydrogen bonds) to a nitrogen atom, what is the name of that compound?

[zsinger]

"Hydrogen bound to Nitrogen, Oxygen, or Fluorine=Hydrogen Bond"-  Thats how we learned it.  Hope that Helps.  Regarding the Amines, Yes, there will usually be a lone pair, unless it has been turned into a quaternary ammonium salt.  Three hydrogens with a nitrogen is ammonia .
          -Zack