I am not sure in this thing.

What is difference between utf8_general_ci and utf8_english_ci collations?

What is better or recommended?

I would use whatever the database defaults to. Trust me - don't mess with db collations and charsets if you can absolutely avoid it lol.

If you're configuring mysql for the first time and want to use one or the other, I use general for any charset groups when possible.

Yes.

I don't have any problems at this time with my database. My tables have set utf8_slovak_ci collation but I don't know if utf8_general_ci would be better.

Changing it would likely result in many problems. I never ever ever ever ever (did I mention, ever) willingly change the collation without VERY good reason once the site is running. It often corrupts the data in the tables.