"I have a bicycle."
Translation:У меня есть велосипед.
есть is the equivalent to the verb "have" (but not its literal translation). In Russian this is a replacement for signaling possession. It does not need further conjugation, as it is a generalized conjugation that can be applied to any present form. If you wanna be specific, then есть would mean "is/are", implying that an object is with you. So, "у меня есть" would literally mean "at me is", and for non-Russian speakers this would mean "I have".
If you don't use "есть", then it will imply that "The bicycle is with me".
«Есть» is not the equivalent to “have”, neither it is an infinitive (its infinitive is «быть»). Originally, it used to be the 3rd person singular form, now is used for all present conjugation. Another surviving form (however found rarely nowadays) is «суть» — which is the 3rd person plural form.
If all infinitives end with ь, it doesn't mean that everything with ь is an infinitive :)
You should use "есть" whenever you state the existence of something (which is otherwise not obvious). On the other hand, you drop it if you simply indicate that you have the object with you (nobody doubts is existence in your posession in general). Hence,
У меня есть велосипед. - I own a bicycle. (There is a bicycle in my possession.)
У меня велосипед. - I have a bicycle with me. (I indicate my mode of transportation.)