A dominant trait means an animal needs only one copy of the allele to display the trait. An example of a completely dominant trait would be black coat color. If an animal gets one copy of the black allele and one copy of the red allele, the animal will be black. A recessive trait means an animal needs to inherit two copies of that allele in order to display the trait (phenotype). This is only possible if both parents have at least one copy of this allele.
For example, red coat color is a recessive trait. A calf must have two copies of the red coat allele (ee) to display a red coat. That does not mean both parents had to be red, they just had to carry at least one copy of the red coat allele. Mating two heterozygous black parents (Ee) would have a red calf (ee) approximately 25% of the time.
Dominance is also important in polygenic traits (traits that involve multiple genes like growth). Dominance is the main reason for hybrid vigor or inbreeding depression as increasing genetic diversity leads to an increase in dominant gene expression and a resulting boost in production.
People often equate a dominant trait to mean “good” and recessive to mean “bad”. While this may frequently be the case (for instance, many lethal mutations are recessive), it is not always the rule. As mentioned about the true definition of dominant or recessive traits has nothing to do with good or bad alleles, just how many copies are needed to display the phenotype. You cannot assume dominant = good and recessive = bad.