It is called a Subframe.
Areas where the subframe bolts to the body are the mounts. There are 2 each rear side for a total of 4, and 2 up front, for a grand total of 6.
Common problems with these subframes is the hole for the subframe donuts will rot off, and the donut goes right through the hole. GM has special plates for these areas, and its a good fix. Another option is to fabricate a new bracket, and just stick it in there.
If it is very severley rotted off, junkyards will have PLENTY of those style subframes. I imagine you can get one dirt cheap, I have no idea. These subframes are the same for the 87-1999 Oldsmobile 98-88, Pontiac Bonneville, and the Buick LeSabre/Park Avenune, from what I know. Its about 8 hours of labour to replace aproximatly.
The difficult thing about the rear bolts. There are 2 per side. They also havea nut that is secured by retainers hidden away. When you screw the bolt in, the retainers secure the hidden nut, without the need to use a wrench or vise-grips. the rear ones are prone to the metal retainers breaking (rust) and the nut spins loose. so you can't un bolt, or put a bolt in. I think you would have to tear into the floor to access this nut to get a pair of vise grips on it. I have to do that tommorrow to my car, lol. Not fun at all.
Anyhoo, hope i was of some help.
2009 Pontiac Vibe AWD 2.4
1993 Buick Park Avenue 3800
1984 Chevrolet Citation 2.5 liter four.