So manual vibrato, as opposed to electronic vibrato, is something that takes a little bit of practice. It's done more on electric guitar than on acoustic guitar, but it might sound something like this.
So that's just me yanking on that string. And it's used in blues probably more than it's used in country, but you've got plenty of country players out there who are always bucking the expectations. Sort of, recasting the mold. You might hear Vince Gill, he probably does some vibrato work in his solos. He's got a real elegant, almost B.B. King style of soloing on the electric guitar on certain songs. You might hear Willie Nelson place some vibrato on his gut string acoustic guitar. You also might want to check out Roy Buchanan. He is the master of the telecaster, at least, he was, and he's got all kinds of crazy, melodic tricks and sound effects going. He's probably got the scariest vibrato of anybody out there.
Vibrato on different strings sounds different. If you're on the, you know, the first string, it's going to be a little less scary than if you're down here on the third or fourth string. That somehow sounds a little spookier. But anyway, it's just a way to dress up your lead work. If you're playing a solo, maybe you're playing a melody, it can be a nice little flourish that takes a left turn where the listener wasn't expecting it to.
One other thing I should mention about it, I like to place three fingers on the single string that I'm using to do the vibrato on. So, say I'm on the first string, I'll have first, second, and third finger all on that first string. And I'm gripping the neck like a baseball bat, kind of the basic grip that I'm often in. And having those two helper fingers behind the finger that I'm using to fret the note really gives me a lot of control, a lot of power. And if I want to do it really fast, that can help me do that. I guess Clapton playing on While My Guitar Gently Weeps, that's pretty much Vibrato 101.
But anyway, just keep at it, use those helper fingers, and you'll be playing a scary vibrato in no time at all.