Navy abbreviations

I'm guessing N. Hamps. and N.H. are New Hampshire. So New Hampshire, Discharged 6/24/1884. Bu. is usually builder.

Does that make sense in the context of the record?

Is this a sailor's discharge record? If so, then "order Bu." would probably be "by order of Bureau of Equipment and Recruiting," which handled personnel matters at the time. 

Records of the Bureau of Naval Personnel

Consulting a few different resources, here is the best answer I've found:

The Bureau of Navigation promulgated the order to discharge the sailor from the ship New Hampshire I or the New Hampshire Rendezvous station in Portsmouth. He may have served as a Fireman 2nd class (Fr.S.), but without more context of the records, this is more of a guess than fact.

You can find out more about the New Hampshire I ship here in the Directory of American Naval Fighting Ships.