Gene is a small segment of DNA that codes for certain proteins which determine the phenotype (external appearance) of a character.

For example, the character of plant height is determined by an height-determining gene.

Alleles are alternative forms of a gene. It controls the expression of a specific trait of the character.  

For example, several traits of the character plant height may include  tall and dwarf plants. These different height of the plant are controlled by the expression of different alleles of the height-determining gene.

Thus, character is determined by expresssion of a gene. And the trait (of the character) is determined by expression of an allele (alternative form of a gene).