Gender agreement in French?

Some nouns, particularly nouns describing humans, have two versions, one masculine and one feminine, in order to match the grammatical gender with the natural gender.

"chanteur" is gramatically masculine. It describes either a male singer or a singer in general.
"chanteuse" is gramatically feminine. It describes a female singer.

An other example : "acteur" is an actor in general or a male actor. "actrice" is a female actor.

The gender agreement is always made with the grammatical gender, not the natural gender. For example, if you are talking about a man and you say "It's a hateful man", you say "C'est un homme haineux", in this case, "homme" is gramatically masculine so you use "haineux" which is masculine. But if you say "It's a hateful person", you would say "C'est une personne haineuse", in this case, "personne" is feminine regardless of the natural gender of the person so you have to use "haineuse" which is feminine.

In order to memorize the gender of a noun, you normally learn it with a determiner. People who are learning French generally learn them with a definite article (the) so instead of just learning "table", "chaise" and "rideau", you learn "la table", "la chaise" or "le rideau".
Native French speakers do not really need to learn a noun with a determiner because learning the gender of a new noun is automatic, however when we do, we use indefinite articles (a/an) so instead of "table", "chaise" and "rideau" we have "une table", "une chaise" and "un rideau".
This has one slight advantage over the other method because it indicates the gender of a noun that starts with a vowel "un albatros", "une innondation" contrary to the definite article method "l'albatros", "l'innondation".

"La chanteuse" is indeed the female form of "le chanteur".

There are a few loose rules how to recognize the gender, but there are many exceptions : http://french.about.com/od/grammar/a/genderpatterns.htm

But the most efficient thing is to always learn each French word together with its correct indefinite article : not "chanteuse", but "une chanteuse".