Monogamy is the practise of only having one sexual parter at the same time. In general, most species are not monogamous. However, there are some species which generally practise monogamy.
Most human cultures practise monogamy, however the practise is seen in other species as well. These include: the lar gibbon, mute swans, the malagasy giant rat, waved albatross, shingleback skink and the convict cichlid.
In many of these cases, the example above is the only species in its order to limit their relationships in this way. For example, the malagasy giant rat is one of only a few monogamous rodents.
Only about 3-5% of mammals bond for life. These include otters, beavers and wolves. Other animals, including penguins, are annually (or seasonally) monogamous.
Within species that are considered monogamous, some individuals can show tendencies to cheat on their partners, or to break off long (established) relationships seemingly without reason - including humans!