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Parallax is a general term used to the way that relative positions of objects appear to change as an observer's viewpoint changes. For example, viewed from a moving train, nearby objects appear to move rapidly past the observer, while more distant landscape features seem to remain relatively static over time. The same is true for stars and other cosmic objects: viewed from changing perspectives, stars that are relatively near to the Sun tend change their apparent position detectably, while those that are much more distant show little or no detectable change. This fact makes it possible to calculate the distance to a star by measuring its parallax angle against fixed background stars.
Measurements of stellar parallax are possible because the Earth is in constant motion around the Sun, completing one orbit in a year. Observations of stars taken six months apart, then, represent views from diametrically opposite points on Earth's orbital path (that is, points two Astronomical Units from one another, the diameter of Earth's orbit).* This baseline is sufficient to show apparent parallax motion in many stars, and from that change in position, a distance can be calculated.
In practice, these apparent changes in position are extremely slight, and are measured in seconds of arc. A second of arc (or arcsecond) represents a sixtieth of an arcminute, and an arcminute represents a sixtieth of a degree, so one arcsecond is equivalent to 1/3600 of a degree. From this is derived a unit of cosmic distance: the theoretical distance to a star with a parallax of one arcsecond is defined as a parallax second (usually abbreviated to parsec). A parsec can be calculated as equivalent to 3.2616 light years, though in fact there are no stars this close to the Sun, which means that all stellar parallax measurements are necessarily smaller than one arcsecond. Typically, these values are expressed in an even smaller angular measurement, the milli-arcsecond (abbreviated mas), which is a thousandth of an arcsecond or 1/3600000 of a degree.
To take an example, the nearest star to the Sun is Proxima Centauri, a red dwarf that actually forms an outlying member of the Alpha Centauri system. Being the nearest star means that Proxima also has the highest known parallax value, measured at 768.0665 mas (or just over three-quarters of an arcsecond). From this, we can calculate a precise distance: Proxima Centauri is 1.3020 parsecs from the Sun, or 4.2465 light years.
Proxima's parallax is exceptionally high, but very few stars are remotely this close to the Solar System. For most stars, parallax values are very considerably smaller (and hence more difficult to measure precisely). For a star one hundred light years from the Sun, the parallax angle would reduce to 32.6 mas, and for a star a thousand light years distant, the angle would be just 3.3 mas.
When the first Earth-based parallax measurements were made in the early nineteenth century, such precise measurements were impossible, but assessment of parallax has improved radically since those early days. Extremely precise measurements are now possible through space-based observations, notably by the Hipparcos and later Gaia missions. Parallax can now be determined with remarkable accuracy, but even so there is a practical limit beyond which this is no longer a useful means of measuring an object's distance. With current techniques, this limit is of the order of about ten thousand light years, but beyond this limit, other techniques of distance measurement need to be employed.
* This baseline of two AU can be extended by taking into account the relative motion of the Sun through the Galaxy, effectively extending the baseline by several AU (depending on the object being observed). This secular parallax technique is more effective for truly distant objects, though the calculation introduces a level of uncertainty that makes the results rather less precise.
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