A trait or character in biology is a feature of a living thing. It is part of an organism's phenotype.

Every living thing, from tiny organisms like bacteria, to plants, animals and humans, has some characteristics which make it special. Thus an elephant has tusks, large size and weight, large ears and very large molar teeth (et cetera). These are typical characters of the African and Indian elephants.

Biologists call those traits. The living thing is built in a certain way; this is its anatomy, its structure or body. The physical structure works in a certain way; this is its function, the way its body works. An animal also acts in a certain way; this is its behavior.

The way that a living thing is structured, the way its body works and the way that it acts are all traits. The basic traits are shared by all the members of the group, that is why they are put in the same group. Other traits are only shared by a small number of the group.

For example: 
 It is an anatomical trait of giraffes to have long necks. All giraffes have this trait. 
 It is a physiological function of birds to lay eggs to produce their young. All birds have this trait. 
 It is part of the behaviour of wolves to live and hunt in packs; it is part of the behaviour of cats to live alone or in small family groups, and to hunt alone. These characteristic behaviours are also traits.

Traits are heritable: they can be passed on from one generation to the next by genes. Mendel's work involved the inheritance of traits on pea plants. The whole group of traits of an organism is its phenotype.

Trait versus character 
Different sources use the term differently. 'Trait' competes with the term 'character'.

Synonyms 
According to some authorities, trait and character are synonyms:
The Dictionary of Genetics: "For trait, see character" and "Character: any detectable phenotypic property of an organism; synonymous with phenotype, trait".

'Trait'  as a sub-character 
Some sources use both terms:
"Any detectable variation of an inherited character. It is the expression of genes as part of the phenotype".

This works well with Mendelian characters. In such cases, a character is a feature of a species which may present as various traits. Examples:
Eye colour = character
Blue eye colour = trait1
Brown eye colour = trait2
Appearance of pea = character
wrinkled = trait1
smooth = trait2

But not all characters are inherited in the simple manner of Mendel's pea characters. For example, an animal's weight is a character, but many genes contribute to it, and so does the animal's environment from birth. Weight is a) continuous rather than discrete (separate steps), b) polygenic (controlled by a number of genes), and c) because weight is influenced by both heredity and environment.

Only 'character' used 
To avoid this issue, some sources use only the term character. Futuyma uses  this system for Mendelian characters:
Eye colour = character
Blue eye colour = character state1
Brown eye colour = character state2
Appearance of pea = character
wrinkled = character state1
smooth = character state2
This use of 'character state' allows Futuyma to use terms such as 'ancestral state' and 'derived state' when talking about the evolution of characters.

References 

Evolutionary biology
Classical genetics