"Cheap" is a word which can be used in any context to mean not expensive or good value.
It can be used colloquially to mean something has been done in poor taste, is of low quality, or has been done unkindly and/or lazily. It's hard to define it easily. A common example is:
"that was a cheap trick / joke!"
"Cheesy" is common in the UK but I don't know about elsewhere. In the UK, it is a colloquial word often used to describe films or TV shows, but it could also be applied to jokes. It means that it was not good quality, perhaps kitsch or sentimental, or lacking in good taste.
Other similar words in the UK are "naff" and "tacky".
Again, "cheesy" is hard to define clearly. It's a word to know but not use unless you've lived in the UK for a while..