Does having an ovarian cyst mean I can't get pregnant right now?

Not at all. Cysts form during ovulation and rupture to then release an egg for potential fertilization and pregnancy. Occasionally they don't rupture and grow too large and cause pain. So discovering a cyst may be just a normal finding, depending on the size. There is a medical condition called polycystic ovary syndrome in which a woman has an imbalance of a female sex hormones that can cause infertility.
No. The ovary is capable of forming many different kinds of cysts. Many of those will not affect a woman's chance of becoming pregnant. So if you don't want to be pregnant, you should use some kind of contraception.