A blue straggler star is a star that is hotter, hence bluer, than the other stars in a star cluster. Blue stragglers, an example of an exotic star, stand out in a cluster because most stars in a cluster are about the same age and color, red. They were first identified by Allen Sandage in 1953 who is best known for his work on the age of the universe and refining the Hubble Constant. To understand a blue straggler and its uniqueness one needs to understand a little about star formation and star clusters.
Stars form from clouds of hydrogen gas and dust condensing until nuclear fusion begins. Stars can form into many different masses. The larger a star's mass the faster it uses its nuclear fuel and the hotter it burns. Stars of less than 1.5 solar masses (Sol has a solar mass of 1) mainly fuse hydrogen into helium and follow a steady path or main sequence eventually becoming a white dwarf. Stars of more than 1.5 solar masses use carbon, nitrogen and oxygen in a process to produce helium from hydrogen. Stars up to ten solar masses can pass through a red giant stage and even more massive stars can explode in supernova or collapse into a black hole. Smaller less dense stars called red dwarfs burn slower and cooler and last much longer than larger more dense stars which burn fast and hot. A hot young star will shine blue, cooler stars, red.
Star clusters generally fall into two groups, open clusters and globular clusters. They form from clouds of hydrogen and dust coalescing into stars all about the same time and tend to stay together either from mutual gravity or inertia. Open clusters or galactic clusters, are sparse star groups containing tens to thousands of stars all formed from the same cloud at the same time and moving through space in the same general direction and speed, they have little motion relative to each other. They stay together because no force is pushing them apart. An example of an open cluster is the constellation called the Pleiades or Seven Sisters, the closest open cluster to the earth and easily visible with the naked eye.
Globular clusters are much more dense and usually much older than open clusters and the number of stars can be many millions. Globular clusters are held together by mutual gravity and tend to form into spheres around their center of gravity. These clusters are very old and the stars that remain from the cluster's creation tend to be red dwarfs, with the larger denser stars having left the main sequence becoming supernova, black holes and the like.
Since globular clusters are majorly made up of red dwarfs, a hot blue star stands out as unique and are called blue stragglers for this reason, they appear to be much younger stars or are straggling behind their neighbors in the main sequence. A blue straggler can be formed in a number of ways. When two less massive stars collide and become one star with increased mass and a hotter temperature. In a binary system, the magnetic drag can slow the orbits of less massive stars allowing them to merge into a single more massive star. A more recent discovery of cannibal or vampire stars that draw the outer shell from a neighboring star in a binary system increasing its mass and adding fuel thus becoming a blue straggler while the looser is not consumed but is reduced to its core and becomes a white dwarf. A blue straggler should show signs of oxygen, carbon and lithium depletion, a chemical signature of mass transfers from older and less dense stars.
For more reading see this article by Aaron Geller.