Learn German: Er hat Geld, aber er kauft kein Haus. - He has money, but he is not buying a house.

hat is the form of the verb haben (to have) used with er (he), sie (she), and es (it). It is slightly irregular: instead of keeping the 'b' from the dictionary form to make 'habt', it drops it completely.

The English translation says 'he is not buying'. Why is there no German word for 'is' in er kauft?

German does not have an '-ing' tense like English does. The single present tense verb kauft translates to both 'buys' and 'is buying'. You never use a translation of 'is' or 'are' combined with another verb to describe an action happening right now.

Why does the sentence use kein instead of nicht to say he isn't buying a house?

When you want to negate a noun that would normally use 'a' or 'an' in English (like 'a house'), German uses kein instead of nicht. You can think of it as literally saying 'he buys no house'.

Doesn't the verb have to be in the second position? Why is kauft the third word after the comma?

The verb is actually in the second position! The word aber (but) is a 'zero-position' word. It connects the two ideas but does not count towards the word order of the second phrase. So er is position one, and kauft sits safely in position two.