What were they for? Exactly what it says on the tin. A "palace of culture" is simply what we would today call a convention centre. A large building where conferences, meetings, concerts, lectures and the like are held. Events such as communist party meetings would be a staple fixture in these places, but their use was not limited to political affairs only, with these often quite massive buildings also housing offices or various leisure venues. They were fairly ubiquitous in the Eastern Bloc, and in many cases, they are used for pretty much the exact same purposes to this day, the architectural blight that is the Prague Congress Centre being just one example.

In addition to "palaces", "houses of culture" also existed. These were basically smaller versions of the above, and once again, were in no way unique to the Soviet sphere of influence. They were community centres or cultural centres just like the ones that have been in use well before the communist regimes established their own or the ones you might find in many cities around the world to this day, provided you account for the particularities of life in a communist country at the time.

I am unaware of any official distinction, at least as far as socialist Czechoslovakia is concerned, between the two. The differences were purely practical, with the smaller venues being more suited to leisure and oriented towards the local community as opposed to hosting international conferences with thousands of attendees. Youth organizations, restaurants, gyms, sports clubs, exhibitions, concert and dance halls, cinemas and the like would be some of the activities or services commonly found in a typical "Kulturní dům" – and in many places, they still are.

Lectures on Marxism-Leninism

No. Lectures on «Political Economy», the «history of the Communist Party», «Scientific Communism» and the «history of philosophy from Marx to Lenin» where given in the educational institutions along with other subjects.

In the houses of culture you had culture. Artemy Troitsky describes how rock music, (first pop rock, then the rest of it,) got slowly accepted enough to be performed in the houses and the palaces of culture in his «Back in the USSR».

He mentions for instance that while, during the Beatles craze, you couldn't have Beatles there you could have Beatles "parodies". Officially they were mocking popular music in the rotting capitalist west but in practice they played Beatles, which everyone loved.

Culture which wasn't accepted yet by the authorities was performed in private flats instead. "kvartirnik" was the term for flat concert and a "kvartirnaya vystavka" is a fine arts exhibition also held in a flat.