The main kind of gun I'm seeing people wanting to ban is an "assault rifle". These are the kinds of guns people would typically use in combat, and are what most militaries have most of their soldiers use. The AK-47 and M16 are probably the two best known assault rifles:

These guns are capable of both "fully automatic" firing where you can hold the trigger to shoot continuously, and "semi-automatic" firing where each pull of the trigger fires once. No additional action, like pulling a handle or pumping a slide, is needed to prepare for another shot. Some guns have settings so they can be fired both fully- and semi-automatically. In general, the US restricts fully-automatic-capable weapons to the police and military, but does allow civilians to own semi-automatics.

The guns used in Orlando (Sig Sauer MCX [1]) and Sandy Hook (Bushmaster M4) were both semi-automatic rifles [2]:

While these two look very militaristic and forbidding, there are functionally equivalent guns like the Ruger Mini-14 that have more of a "rancher and hunter" aesthetic:

Or, if configured to hold 20 bullets instead of 5:

We did try to ban guns like this before, in 1994, with an Assault Weapons Ban that tried to ban the more military-like semi-automatics while still allowing others. It did this by banning a list of models by name, and then didn't allow others to have more than one of a list of features. This mostly didn't work, though, and a DoJ review determined that "the [Assault Weapon] provision targets a relatively small number of weapons based on features that have little to do with the weapons' operation" (Koper 2004, pdf).

If we want a kind of gun restriction that's helpful for preventing mass shootings, we want to reduce the number of people that can be killed in a short time period, especially before the shooter has to take time to reload or switch guns. So one thing that could work would be limiting the capacity of magazines (the bit that holds bullets) to something like five shots. Look at the Mini-14 pictures above and it's not a coincidence that the lower one looks more menacing: it can shoot four times as many people! Ideally you would ban high capacity magazines entirely; there are enough out there already that just prohibiting selling new ones wouldn't do much. The advantage of this politically is that for most non-combat non-mass-shooting uses, you just don't need that many sequential shots. So there's some chance of getting both sides to meet here.

On the other hand, mass shootings are a small fraction of gun homicides, so specifically limiting guns that are well suited for mass shooting might just reduce the fraction of gun deaths that are newsworthy without reducting overall deaths by very much.

(This is all very complicated, and I'm still not as informed as I'd like to be.)

[1] Not an AR-15 as initially reported.

[2] Technically the M4 is a carbine, and was designed as a somewhat shorter and lighter version of the M16 rifle.