In the chapter before the LORD had even spoken the ten commandments, and a few chapters before describing how to build the tabernacle, the LORD mentioned these other priests:

Exodus 19:20 And the LORD came down upon mount Sinai, on the top of the mount: and the LORD called Moses up to the top of the mount; and Moses went up.

  21 And the LORD said unto Moses, Go down, charge the people, lest they break through unto the LORD to gaze, and many of them perish.

  22 And let the priests also, which come near to the LORD, sanctify themselves, lest the LORD break forth upon them.

  23 And Moses said unto the LORD, The people cannot come up to mount Sinai: for thou chargedst us, saying, Set bounds about the mount, and sanctify it.

  24 And the LORD said unto him, Away, get thee down, and thou shalt come up, thou, and Aaron with thee: but let not the priests and the people break through to come up unto the LORD, lest he break forth upon them.

Who were those other priests, before the LORD described how Aaron and his sons would be priests?

Exodus 24:4 And Moses wrote all the words of the LORD, and rose up early in the morning, and builded an altar under the hill, and twelve pillars, according to the twelve tribes of Israel.

  5 And he sent young men of the children of Israel, which offered burnt offerings, and sacrificed peace offerings of oxen unto the LORD.

Were those other priests possibly the young men which offered burnt offerings, mentioned in Exodus 24:5, shown above?

It wasn't until chapter 28, that Aaron and his sons were specifically declared to be the priests in Israel:

Exodus 28:1 And take thou unto thee Aaron thy brother, and his sons with him, from among the children of Israel, that he may minister unto me in the priest’s office, even Aaron, Nadab and Abihu, Eleazar and Ithamar, Aaron’s sons.