Almost every Ithaca pump you see will be either a model 37 or 87. They are nearly identical. The model 37 is pre-bankruptcy and is chambered for 2 3/4" (in 12 guage) and the model 87 is post bankruptcy, chambered for 3". They've been manufactured for over 60 years, and come in about every style and grade you can imagine.
I'll neglect the hunting versions, although my father-in-law borrowed one once for an upland hunt, and liked it so much he almost bought one.
For serious defensive work, there are 5 and 8 round versions. (4+1 and 7+1). The one you show is the 5-round, with pistol grip available both from the factory and aftermarket). The barrel is very short, under 18". I think that version (short barrel, pistol grip) is known as a model 37 Stakeout. Since the GCA of 1968, the super short barreled variant has, of course, been a class III weapon. I think the BATF grandfathered pre-1968 versions since they were the only commercially produced shotgun with that short a barrel. If you owned one, you still had to register it, but you got a one-time exemption from the tax. I'm not too sure on the details.
Mine is an 8-shot M&P (Military and Police) model 37 with a 20 inch barrel. I've never heard of an 8 shot model 37 in any other barrel length. It would be foolish to saw that off even without the legal hassle, since the magazine is nearly as long as the barrel! There is also a DSPS model which was much more popular than the M&P. It has a large front sight (mine has just a bead), and the barrel is bored for slugs - a little tighter than normal over the entire length of the barrel, and no choke whatsoever. This gives it a little more accuracy for firing slugs. DSPS means "Deer Slayer (or is that Stalker?) Police Special". There is also a long-barrel DS model that was developed for - you guessed it - hunting deer with slugs.
The model 37 was designed by John Browning for Remington. I think they produced a version (Remington model 17 or 20...don't have my reference books handy!) for a few years, then sold the rights to Ithaca. Remington then started manufacturing a new pumpgun, you might know it as the model 870...
Ithaca took Browning's design to the edge of 1930's technology, milling the entire receiver from a single block of metal in order to save weight. To this day, it's still the lightest pumpgun in any given gage, magazine capacity, and barrel length. Most of them were sold as "Model 37 Featherweights" (or Featherweight DSPS, Featherweight Stakeout, etc.) A typical 12-gage model 37 weighs 6 1/2 pounds, less for the Stakeout. One variant in 20-gage was known as the "Ultraweight".
The Model 37/87's real claim to fame is the speed of it's action. It has a shorter slide throw than any other pumpgun, allowing a skilled practicioner to easily fire - oh, 5 rounds in just under 2.0 seconds. For this reason, it is the official "riot gun" for many police departments, including the NYPD and the LAPD. An oddity of the firearm that has some combat value is the lack of a disconnector. This means that if you hold the trigger down while cycling the action, the weapon will fire as soon as the slide reaches the full forward limit of travel, without need for releasing and pulling the trigger for each shot. This is a characteristic shared with the Winchester model 12 and model 97.
I'll close with a description of a trick shot that professionals have performed for the amusement of paying audiences. It takes a shotgun without a disconnector, such as the Ithaca 37/87 or a Winchester 12, a coffee can, and some flour. You load the firearm, put an unspecified quantity of flour in the coffee can, and toss it high overhead. Just as it reaches the apex of its arc, you shoot it as quickly as you can, as many times as you can. The flour makes a big white puff every time you hit it (proof of your marksmanship to the onlookers). The can flies faster and higher with every shot, and an excellant wingshot can put 5 of 5 into the can before it sails out of range. Rock and roll!