Bonds in R-H3N+ group Source: Wikipedia

I am confused about the ammonium group in this molecule.

I can see that we have a nitrogen atom and 3 hydrogens. The group is attached to a carbon.

Nitrogen has atomic number 7 so my understanding is it can form up to 3 covalent bonds, and therefore cannot have covalent bonds with the carbon and all 3 hydrogens.

I can see a + on the nitrogen, suggesting it is an ion, and maybe some ionic bond?
Nitrogen along with it's 3 bonds has a lone pair. That's what comes into play. The H+ ion has no electrons of it's own and therefore when it forms bond with N, nitrogen coordinates it's lone pair to H+ completing it's duplet and at the same time it's octet and forms a coordinate bond also known as dative-bond.

You can find more about coordinate bond here