That happens to all of us who tie flies at one time or another. So what gives? Well, it's most likely that when it's happened to me, its because I've been in a hurry to get a fly started & didn't mount the hook in the jaws of my vise correctly. That's usually why a hook slips in the vise jaws.
If you've mounted your hook correctly in the vise & it still keeps slipping, it might be that your vise jaws are dirty. Take them apart & give them a good cleaning. While you're doing that check to see the condition of the jaws. If they're worn out then you'll need new ones, but this is kinda rare as a good vise should last you a lifetime.
If you're new to tying flies, or just keep having your hooks slip in the vise jaws, then take a quick look at the short video below. It will show you the proper way to mount a hook into any fly tying vise - not just the vise they're showing.
I hate it when my hook slips from the vise & if I'd only take a moment to make sure I've put it in the vise like in the video below, I wouldn't have any trouble.