Rockwell was the manufacturer.

B-1B is the aircraft type designator:

• B=Bomber
• 1=1st serial (numbers were restarted at 1 just
  before the B-1 was designed)
• B=Second variant. A=1, B=2, C=3...

I beleive Boeing bought out Rockwell, and inherited everything from it.

Currently P no longer stands for Fighter, it now stands for Patrol. Also the Avro Arrow could not turn at the rate a F-22 can, it can out gun and out manuvear almost any aircraft in the world.

P.S. F-14's ALL THE WAY 

P was Pursuit.

Canada is not the only country to add a national identifier.

Other that I have heard:

NF Norway, Fighter
EF Spanish. Fighter

They may be others...

I believe it did.  If I remember right, that was given to the fighters because their primary planned use (at least really early on) was to chase enemy planes and shoot them down, they wern't for attack but defense (basically interceptors).  At least I read that somewhere-I could be wrong.  Eventually planes like the P-47 were build as fighter/bombers and the persuit designation no longer fit.

  At one time Rockwell was affiliated with North American, maker of the P-51, F-86 and F-100.  This was probably Rockwells entry into the aerospace business.  Rockwell also did work on the X-15, I believe the X-24 Lifting Body and the Space Shuttle.
  In recent years, with the shrinking aerospace market, many of the smaller aerospace companies were absorbed by larger ones leaving only three major military aircraft manufactures in the U.S.:  Boeing, Lockheed-Martin and Northrop-Grumman.
  The shrinking areospace market has made each military contract a "do or die" situation for these companies and they have become very competitive with each other, as seen in the recent JSF and UCAV competitions.

  As to aircraft designation, before 1947 the Air Corps was a division of the U.S. Army and under the command of the U.S. Army.  During this period fighter aircraft received the designation of "P for Pursuit".  In 1947 the Air Corps was separated from the U.S. Army and become its own identity, the U.S. Air Force.  At that time the "P for Pursuit" was changed to "F for Fightger", P-51s and P-80s became F-51s and F-80s.  Now the "P" designation is used for "Patrol", as in P-3 Orion.   

  So, why are there no B-75s or F-121s.  Well, back in the mid 1960s there was a move to simplify aircraft designations for book keeping reasons.  At that time the Air Force and Navy used different numbering systems to designate aircraft. This was considered to be confusing, wasteful and a redundant copy of paper work.  To remedy this no further "Century" aircraft numbers were issued to designate Air Force aircraft, the F-111 being the last.  Also, on the Navy side, the manufacturer's letter was dropped from their designations, thus the F8U Crusader became the F-8 Crusader and the F4H Phantom became the F-4 Phantom.  This progression continued with the F-14 through to the F-22.
  As for bombers, it was simpler to start over with B-1 after the B-70.      

Yeah, Boeing bought out Rockwell.