Beszél means - speak/sprechen, reden (there is often an aim to "beszél" with someone, it often implies a more serious talk about a certain topic)
Beszélget means - chat/sich unterhalten, plaudern (it's often a (longer) dialogue between friends, acquaintances etc about various topics.)
Examples for "beszél":
A tanár földrajzórán a tengerekről beszélt. (The teacher spoke about the seas in Geography lesson).
Beszélni fogok a főnökömmel! (I will speak/talk to my boss).
Examples for "Beszélget"
Tegnap órákon át beszélgettem a barátnőmmel. ( I was chatting/talking to my girlfriend for hours yesterday.)
Jó lenne megint beszélgetni! (It would be good to chat/talk again!)