공부하다 is the infinitive. 공부한다 is its declarative conjugation.

공부하다 is the base form (or "root", "dictionary", or "infinitive" form) of the verb. This form is not used in normal conversation or writing except in certain obscure contexts. 공부한다 is one of the four declarative ending forms for the present tense, which are the most common in usage. The four are: 공부합니다(formal-polite), 공부한다(formal-casual), 공부해요(informal-polite), 공부해(informal-casual). Every verb can take these basic endings and the other lesser ones(-지, -네, -ㄴ/는군, etc), plus the "connective" forms. Here are some examples (basic endings / less used endings / connective forms): * 가다: 갑니다, 간다, 가요, 가 / 가지, 가네, 가리, 가는군, ... / 가서, 가고, 가도, 가면, 가더라도, etc. * 크다: 큽니다, 크다, 커요, 커 / 크지, 크네, 크리, 크군, ... / 커서, 크고, 커도, 크면, 크더라도, etc. You can check how a particular verb conjugates at dongsa.net (it shows most but not all). Type in the base form in the "Search" window, then it will show about 43 different conjugational forms.