When you see the word Hyper in relation to space, you might think of Hyperspace, the faster than light travel that vehicles such as the Millennium Falcon do in Star Wars. Sadly, the vast majority of Hypervelocity stars are nowhere near as fast. There are galaxies that are said to be moving at the speed of light though which we will get onto later.
All stars move including our star, the Sun orbit the Galactic Centre of the milky way which is located in the constellation of Sagittarius the Archer on the borders with Scorpius, the Scorpion. If you look up at the sky at night, you won't notice a thing, all the stars will be static in your eyes. If you take a photo of the night sky with a slow exposure, the stars will have trails because they've moved.
There aren't that many hypervelocity stars so far discovered, in fact they were only discovered about 10 years ago so their field of investigation is quite new. Before then Jack Hills of Los Alamos National Laboratory in U.S.A. had theorized these stars in 1998 but it took more years to be discovered. The first was discovered in 2005 by Warren Brown and the star in question was SDSS J090745.0+024507 in the constellation of Hydra. Warren's star has an escape velocity of 853 +/- 12 km/s, B.B.C.
The majority of the Hypervelocity stars are presumed to be stars that were in a Multiple Star System such as binary or more. The group was orbiting close to the Galactic Centre when it gets too close. One of the stars in the system is sucked into the centre and the others are ejected. The stars are then forced to head towards the exit point of the galaxy and become a rogue star.
The Sun is currently moving around the Galactic Centre at a speed of 828,000 km/hr which might seem a lot but when you consider how far it has to travel, it'll seem like nothing. Ref: N.A.S.A. It is located at approx 28,000 light years from the Galactic Centre and we're in the suburbs. Some stars such as AH Scorpii will take less time as its closer to the Centre whereas Elnath will take longer. Elnath in the constellation of Taurus is the closest major star to the Galactic Anti-Centre.
Hypervelocity stars are discovered using the doppler effect. If a driver of a car has his radio on at a constant volume, as it comes near you, it gets louder and as it moves away, it gets quieter even though the driver hasn't changed the volume control. The same is with stars, a stellar doppler effect is applied using the spectrum of the light from the star. If the star has more red then its moving away, if its moving closer then its blue. By analyzing the spectrum, you can determine how fast it is moving.
There are some exceptions to the origins of a Hypervelocity star, star US 708 which has the record for fastest velocity as at 2015 doesn't come from the Galactic Centre. It is believed the star was in a binary star system and the other star was orbiting it very fast. The other star in the system was a white dwarf star, the remnants of a dead star.
When 708 exhausted its hydrogen, it engulfed the white dwarf which was still orbiting and 708 turned into a sub-dwarf star. The white dwarf seemed to be sucking helium from the larger star, a vampire star. When the white dwarf became too unstable, it exploded and blew 708 away causing it to become a Hypervelocity star.
In the constellation of Ophiuchus is the star Zeta Ophiuchi also known as Han, a star that is visible with the naked eye. You won't be able to see it move but unlike other Hypervelocity stars, it is one that you can see with the naked eye. All if not most other hypervelocity stars are too dim to see from Earth and have been detected using telescopes.
The Zeta Ophiuchi is a star that is 20 times more massive than the Sun and therefore won't live as long. Zeta Ophiuchi is the small blue star in the middle of the dust cloud. It is believed to be the same type of hypervelocity star as US708, its velocity caused by a companion star exploding and forcing Zeta Ophiuchi to move.
The blue main sequence star HE 0437-5439 may actually be a hypervelocity star from another galaxy, the Large Magellanic Cloud galaxy to be more precise. The Large Magellanic Cloud is a satellite galaxy to our own Milky Way. The star's speed has been measured at 1,600,000 miles per hour (2,600,000 km/h) which is three times faster than the Sun orbiting the galactic centre. ref: E.S.O..
Part of the reason why it is believed to be an alien star is because of its relatively young age and the distance it has travelled already. The elements that make up the star are more akin to those in the Large Magellanic Cloud. Ref: Space Daily
Proper Motion which you may have heard about is the angular velocity of an object as it moves across the sky. Only nearby stars can be measured for Proper Motion because we can detect their movement across the sky. The star with the highest proper motion is Barnard's Star in the constellation of Ophiuchus. It moves at about 10328.12 milliarcseconds declination and -798.58 milliarcseconds right ascension. There could be star with a larger proper motion but we not discovered it yet.
Proper Motion is not therefore the measure of a star as it moves through the cosmos. It is only how it moves across the sky in relation to us here on Earth. High Proper Motion stars aren't classed as hypervelocity stars.