The tribe of Levi, led by Moses and Aaron, had the responsibility to keep all of the instruments of the Tabernacle. And, just like all of the other tribes, they had a prince to lead them; Aaron was their leader.
Now, when God had Moses count the armies of all of the other tribes of Israel, they were numbered from the age of 20, and up. But when God had the Levites counted, He included all of the males from one month old and up. Of course, they weren't warriors ... they were worship leaders.
Now, the tribe of Levi was significantly smaller than all of the other tribes. And, Levi was segmented into 3 groups according to the names of Levi's 3 sons (Genesis 46:11); Gershon, Kohath and Merari. And, the Levites camped (by families) immediately around the Tabernacle. The whole tribe of Levi was only 22,000, or 22,300 if you add the families as they were listed previously (Numbers 3:39). Matthew Henry attributes the discrepancy of 300 to the fact that some of the Levites were firstborn themselves, and so, though they were counted in their families they were not counted in the total comparison of Levites to firstborn Israelites since they certainly could not be counted twice (for themselves and for others). This fits well because the very theme of this section relates to that fact. God did indeed claim the tribe of Levi for Himself in place of all of the firstborn males in the whole nation... which, at that time amounted to 22,273 (Numbers 3:43). In fact, since there were 273 more firstborn sons in Israel than there were Levites to substitute for them, God required a payment from the 22,273 eldest Israelites to make up the difference.