What is the parallax angle?
The parallax angle is the angle between the Earth at one time of year, and the Earth six months later, as measured from a nearby star. Astronomers use this angle to find the distance from the Earth to that star.
The Earth revolves around the Sun every year, so that every half year (six months) it is on the opposite side of the sun from where it was six months ago. Because of this, nearby stars will seem to move relative to distant, "background" stars. You can see this effect driving in the country. Best way to see this by holding a thumb at arm's length relative to some background (a painting in the wall a chair in front of you whatever works) and look at it through one eye, then the other. Notice how it shifts position, yet the your thumb hasn't actually moved. Your eyes model the different positions the Earth is in, first on one side of the Sun (your nose), then the other.
Astronomers look at the sky on a specific date, and then six months later, to see how far a nearby star appears to move relative to the background. The angle these astronomers measure the star to move is actually the same angle they would see the Earth move if they could travel to the star. Because the scientists know the distance the Earth has traveled in six months (twice the distance to the Sun), they have all the information they need to find the distance to the star.