Kel tec PF-9, or Kahr PM-9. The way i see things for a carry pistol, if it aint a full on 9mm or larger caliber, ill stick with a 22lr with capacity and low recoil. There's a good chance that's just me though.
My primary carry gun is concealed so a backup for me would probably be concealed or probably deeply concealed. That means for a backup I'd be looking at something like the Ruger LCR or maybe the micro-compact Kahr MK9.
If I were carrying an unconcealed primary handgun I might be able to get away with a larger backup--something like the Kahr P9 or another subcompact single-stack 9mm.
If I were carrying a rifle as my primary firearm then I would likely carry an unconcealed handgun as a backup. I'd pick something like a full-sized, high-capacity pistol in the Beretta 9X/Full-sized Glock/CZ75/H&K USP/Beretta PX4 size range.
If i really felt the need for a "back up gun" in a social carry sense, i'd just carry another pf-9 in my other pocket. They are potent enough to be a primary carry, and small enough for anything you'd need an easily and deeply concealed gun for. Plus you can get two Keltec PF-9's for the price of one khar PM9.
If we are talking about concealment, as a Backup...Sometimes it's a 5 shot .357, sometimes it's a Glock 27. Sometimes I don't even carry a larger handgun, and I just carry both of those.
A consideration should also be the caliber. I find it advantageous to have the BUG the same caliber as the primary. I always carry two spare, full magazines. If, hypothetically, my Primary fails and I have to used the BUG, I have more ammo by emptying what I need from those spare mags. Otherwise, one may be inclined to carry another set of spare mags filled with a different caliber if the gun were different.
If you mean a true backup, as in you wish to carry two weapons, more than likely you will looking for something on an ankle holster. For pure comfort and fit, most compact or subcompacts in a .380 are comfortable. I wouldn't go any bigger than that for a backup. Like a previous post listing the CZ-83, the CZ82 isn't bad either (9mm Makarov).
I generally only carry a BUG when I'm on duty, however there are times that I'll carry two guns when off duty. Mine is a S&W 360J scandium j-frame that has been rechambered to 9x23 and had some other work done on it. I usually carry it loaded with Federal 9BPLE 115grJHP +P+ 9mm's, my current duty gun is a 9mm Beretta Cougar loaded with the same ammo.
I do occcasionally carry a .38 Super 1911 loaded with either .38 Super or 9x23 Winchester, when I carry that I load my BUG with the same ammo as the 1911.
This gun has worked out very well for me thanks to it's versatility, size, and lightweight (13oz). I generally carry it in my off hand cargo pocket. The pants I wear have a second pocket sewn to the pant leg inside the cargo pocket and it rides there perfectly. It's well hidden but secure and easy to access quickly.
Keltec P-11 9mm or one of the newer Keltec's. I picked Keltec as I have a P-11 and it has fed every hollow point I have put through it with 100% reliability and is accurate, small and lightweight.
I see this has morphed into more of a " I carry this as a BUG" so I will chime in and say that I carry an NAA Mini 22 Mag to back up a 38 5 shot snub. I also use the same 5 shot snub to back up an HK P7 on occasion. I never carry 3 guns. I rarely carry 2.
I think the ultimate BUG would be the S&W 342 PD sans the lock.