The dire weasel (MM 65) and the weasel (MM 282) have the special attack attach, which says
If [the creature] hits with a bite attack, it uses its powerful jaws to latch onto the opponent’s body and automatically deals bite damage each round it remains attached. An attached weasel loses its Dexterity bonus to Armor Class….
An attached [creature] can be struck with a weapon or grappled itself. To remove an attached weasel through grappling, the opponent must achieve a pin against the creature.
Unlike the stirge (MM 236-7)—which has an identically-named special attack—, the weasels' attach mentions nothing about the weasels grappling. While it's clear the Monster Manual's authors had some kind of vision as to how this special attack should work, how does this special attack work in actual play? That is, can an opponent with a weasel attached to him simply move away to free himself from the "powerful jaws" of the weasel? Does an attached weasel instead immobilize its opponent? Does the opponent's movement take the weasel with it? Is the weasel actually grappling—moving into its opponent's square, making grapple checks, and so on—while it's attached?
Note: In nearly two decades of playing this game, I've never used a weasel, dire or otherwise. Now one of the PCs has a weasel familiar, and he'll likely buff it and send it in to drain his foes' blood, and I'm gonna need to know how that works. An outstanding answer could go so far as to walk me through a couple of rounds of combat between, for example, a generic level 1 commoner and a weasel.
Best Answer
Weasels, in real life, lock their jaws when they bite down. That's what this is modeling - it bites, and now its jaws are locked and it's attached to you. It doesn't have to make any additional checks to stay attached, grapple or otherwise. It is "grapple-like" in that it's attached to you and loses its Dex bonus, but other things (grappling/grappled conditions, etc.) don't apply.
You can move, but it doesn't unattach the weasel. If it's just a normal sized weasel, it's probably pretty easy to move with it attached to you. If it's giant, then it's probably not so easy; you would need to do a grapple check yourself and use the "move" option. It is not technically grappling you, so they don't say you are grappled. It doesn't move into your square (unless it was a Tiny one and had to in order to attack you).
The way to unattach the jaws is use grappling yourself to achieve a pin (and unlock the jaws). Or kill it, though technically in the real world this doesn't get the weasel's jaws unlocked, so it would be entertaining to make them go through more work to get it off them.
I had a weasel familiar in one campaign but using it to kill downed opponents was about all it was safe to do; because when it bites it's only doing like 1 hp/round and can easily be beaten to death by most any opponent with its newly lowered AC.
In e.g. AD&D 2e the weasels have more description that makes it clear that they are biting down and continuing to suck blood. This means that it's clear the weasel isn't supposed to be able to attach to multiple opponents. As this question isn't tagged [rules-as-written] I see no reason to go past common sense to "well it doesn't SAY it can't do that" lengths.
If you would like to make this easier, the Pathfinder giant/dire weasel just has "grab" and then the blood drain, so it works more within the rules.