Depending on how stars resemble the Sun, astronomers refer to them as "solar type" stars, "solar analogs" and "solar twins". Solar twins are the the most similar to the Sun.
A solar type star actually belongs to a broad category. Back in the XIX century, Father Angelo Secchi (1868, Sugli Spettri Prismatici Delle Stelle Fisse) observed bright stars using a spectrograph (an instrument that splits the light with such a power than besides the colours we can see lines from different chemical elements) and found that yellow stars showed a spectrum similar to the Sun's, hence called them tipo solare. The similarity was due to the limitations of early spectrographs. Work performed later divided those yellow stars actually in 3 different types, F, G (like the Sun) and K stars (see more about Stellar classification here), More specifically, solar type stars include stars from late F (the coolest F stars), G (like our Sun) and early K (the hottest K stars).
A solar analog is more similar to the Sun, yet with important variations, such as differences in temperatures of up to 250 degrees in relation to the Sun, and with a chemical composition within a factor of 2 - 3 of the Sun's.
Solar twins are stars nearly identical to the Sun in all their properties (such as mass, temperature, chemical composition). For example, it has been suggested that solar twins should have temperatures within 100 degrees and chemical composition within 25% of the Sun (Ramírez 2009). The first star recognised as a solar twin was 18 Sco. This is the brightest solar twin, yet it is barely visible to the naked eye. The star that most approaches the concept of solar twin is HIP 56948, also known as Intipa Awachan. Since stars of the same mass and chemical composition evolve in the same way, astronomers also call solar twins those stars with about the Sun's mass and composition. Hence, solar twins evolve as the Sun. Solar twins of different ages gives us the fascinating possibility of studying the past and future of the Sun. The oldest known solar twin is HIP 102152.