What makes breakers different from each other?
The way breakers fit, hook and lock onto the bus is unique for each panel type.  They must not be interchanged. This isn't about brand loyalty, it's about physical fit and safe performance.  If Eaton really wanted to, they could make breakers for QO panels. They would have to have UL test, list, and label them as for QO.  Often you can physically force type X to clip onto type Y's bus... but it will not perform reliably, and it's not allowed under 110.3. 

Because of the need to have people not put CH breakers in QO panels, they get different names - CH and QO.  Sorry if saying that is a bit obvious. 

When you're buying panels, that means price not only the panel but the breakers you'll need for the panel, especially the oddballs like AFCI, GFCI, double-stuff, etc.  (Though I hope anyone who graduates from the Tester-TPE-Harper School of Speccing Panels never has call for a double-stuff.)

So your general families are: 


Eaton CH - commercial grade with 3/4" wide breakers.  CH is for Cutler-Hammer's original line.  Eaton acquired Cutler-Hammer. 
Eaton BR - cheapie builder/home grade with 1" wide breakers.  BR is for BRyant, which Cutler Hammer acquired.  Of late, Eaton has been starting to use CH "guts" in BR breakers, which can only be good for quality. 
Square D  QO, Square D's  line of commercial-grade breakers. These are 3/4" wide. QO is for "Quik-Open".  They also make a variation of QO where the breakers bolt down, a feature desired in industry.  This may be something you want for a generator-interlocked panel, since backfed breakers must be bolted down. 
Square D Homeline - truth in advertising.  1" wide.   
GE Qline - an industrial grade family of standard 1" panels. However they have a very strange take on "Double-stuff" breakers - certain busbars have an extra bit, and they sell 1/2" wide breakers that can clip onto that bit.  2-pole breakers are 1" wide and straddle 2 spaces. 
Siemens Q -- an industrial grade, like QO they make variations that "snap in" or that "bolt down", again of interest for backfed breakers.
Siemens Murray - another quality line inherited by Siemens. 


There are a few others, let me know in comments if I missed one, or just edit it in.  Now on to obsolete panels. 


Pushmatic -- a slick system that is fine, just obsolete.  Pushmatic had 2 columns of square-ish breakers that bolt down. They had a simple bus-bar left and right (L1 and L2), and didn't bother with the comb-meshing of modern panels. A 2-pole breaker was double width instead of double-height. Breakers are right next to each other so handle-tying is easy.  Breakers are still made, even double-stuff, but GFCI/AFCI are nonexistent - use an external deadface. 
Federal Pacific  (FPE) -- Canadian version okay, US VERSION UNSAFE. Kill it with fire before it kills you with fire.  Like VW, they cheated on their testing. While the breakers are defective, the panel's bus bars are also defective and cannot be made safe.  Eaton makes kits to completely replace the guts of a FPE panel, reusing only the shell, which means you don't have to pull all the cables out of it, that being the bulk of the labor in a panel change.
Zinsco has the same problem as FPE. 


Zinsco panels have a neat feature that can really getcha when you convert.  Each breaker fits over both bars, and you can flip the clip so it grabs either bar. On their double-stuff 1-pole breakers, you could grab 1 of each pole.  That's perfect for MWBCs.  Trouble is, if you move it "just like that" over to a modern double-stuff, both MWBC hots will be on the same pole - overloading the neutral, as this guy did in this video.  (of course he's using a GE Qline, so all he has to do is move the breaker up 1/2 space). 

Classified breakers

Now, there is a legal process by which brand X can make breakers specifically for Brand Y panels. This is called a UL-classified breaker.  (I suppose other NRTLs could classify breakers, but all eyes look toward UL on this subject).  Square D refuses to have anything to do with this, and rejects the very concept of classified breakers.  


Eaton CHQ is classified for Square D "QO" panels.
Siemens QD and MSQ are also classified for Square D "QO" panels. 
The Eaton CL line (which is NOT CH or BR) works in a variety of panels, including Siemens Q and Murray; GE, Square D Homeline, and several obsolescent types such as Crouse-Hinds and T&amp;B. CL does not fit Eaton or Challenger panels. 
The Eaton CL line includes some neat breakers that don't really exist natively, such as their remote control breakers, and AFCI or GFCI for the obsolete panels they fit.  
The Eaton BR line is actually UL-listed as dual Type BR and C, meaning it is natively listed (not classified) for Challenger panels.  BR is not CL and should not be put in any other panel. 


Note that classified breakers by Brand X do not fit any Brand X panels. For instance Eaton CL is not Eaton BR.  Do not put a CL breaker where a BR belongs.