Strictly speaking, only the Tribal Assembly (comitia tributa) could pass a law (lex). This assembly was made of the entire Roman people (free males, of course) divided into their 35 tribes, with each tribe having one vote. A magistrate (consul for sure. Tribune of the Plebs also?) would announce a law and a set number of days later the tribes would vote, with a majority of the voting blocks carrying the day.

Traditionally any law proposed first had to clear the Senate, but this was only tradition and in fact when the Senate would not approve Caesar's legislation, he took it directly to the tribes. This might be considered the source of the conflict that led to the civil war, though I wouldn't take it that far. But for various reasons Caesar's enemies considered his legislation illegal, and securing his legislation did motivate a lot of Caesar's actions going into the civil war.

A subset of this assembly is the Plebeian Assembly, which is the assembly of the plebs - patricians were excluded. This was summoned by Plebeian Tribunes and could pass what's called a "plebiscite" which over time became to be respected as having the force of law, but it's not technically a lex.