We can define the parallax as the apparent change in the position of an object due to a change in the location of the observer. What you need in order to measure the parallax is the observation of a distant object from two vantage points, the distance between the two vantage points, and a measurement of an angle.
Consider this simple measurement. Stretch your right arm and close the left eye. With your right eye open, position your thumb in the line of sight of a distant object. Keeping the arm in the same outstretched position, open the left eye and close the right one. You'll see that, with respect to the outstretched arm, the position of the far waway object has shifted. By measuring the angle by which it has shifted, one can get a measurement of the distance. The further apart are the observation points, the better the measurement is.
In astronomy, parallax measurements take advantage of the fact that, as the Earth orbits around the Sun, relatively near-by stars appear to move with respect to the fixed, very distant stars (see the diagram below).
In order to measure the parallax of objects which are very far away from us, we have to use the largest baseline possible (The baseline is the distance between the two points where we take the measurements). Of course the largest baseline available to us for ground based observations is the diameter of our planet's orbit. Using the Earth's orbit, we make one measurement of the position of a star at two times half a year apart, for example on March 31st and on September 31.
The smallest shift we can measure from the Earth is 0.02 seconds of arc, or arc-seconds, which corresponds to a distance of about 50 parsecs (The parsec is the distance a star would have if its parallax is 1 second of arc. - a degree is divided in 60 arc-minutes and an arc-minute is divided in 60 arc-seconds -
1 parsec = 206,265 AU ~ 3.26 light-years. Often distances are given in kiloparsecs - 103 parsecs - and megaparsecs - 106 parsecs).
Parallax measurements of stars are limited by how accurately one can measure small angles. The farther a star is, the smaller the parallax angle. If a star is farther than 50 parsecs it will not appear to move with respect to the fixed background stars we use in the measurement. a-Centauri, our closest star, has a parallax of only 0.76 seconds of arc.
Let us write down some useful formulae (see also the figure above):
1) Star parallax relation: parallax angle
2p (in arc seconds) = 2 B / d.The baseline 2 B is in units of AU and the distance d is in units of parsecs. For measurements from the Earth, B = 1 AU.
For example, a star 2 parsecs away has an arc p of:
p = 1 / 2 = 0.5 arc-seconds
2) Star distance
d = B / p.
For example, if p=0.76 arc-seconds, B=1 AU, then d=1.3 parsecs = 4.3 light-years, since 1 parsec = 3.26 light-years.
Remember: parallax angle is an angle and is measured in radians. Parsec is a unit of distance.
In summary: if you work with the Solar System, use AU; if you work with stars and galaxies use light-year(ly), parsec(pc) or kiloparsec(kpc). For distances to other galaxies, use millions of light years or megaparsec (Mpc).