French has two verbs which can be translated by the English verb to know: savoir and connaître. This might seem confusing to English speakers, but in fact there are distinct differences in meaning and usage for the two verbs.
Meaning
Savoir means to know
a fact
by heart
how to do something
Connaître means
to know (someone)
to be familiar with (someone or something)
In the passé composé
Savoir means to learn or find out.
Connaître means to meet someone for the first time.
Ils ont su que j'ai gagné.
They found out that I won.
Elle a connu Luc à Rome.
She met Luc in Rome.

Grammar
Savoir is often followed by an infinitive or a subordinate clause.
Connaître is always followed by a direct object.
Je sais où il est.
I know where he is.
Je connais bien ton père.
I know your father well.
Il sait le poème (par coeur).
He knows the poem (by heart).
Je connais son poème.
I am familiar with his poem.
Nous savons nager.
We know how to swim.
Nous connaissons Paris.
We know/are familiar with Paris.

Conjugations
Savoir and connaître are both irregular verbs.
Savoir conjugations
Connaître conjugations

Ignorer is a related verb which means not to know in the sense of to be unaware of. Depending on the context, it can replace either ne pas savoir or ne pas connaître.
J'ignore quand il arrivera.
I don't know when he is arriving.
Il ignore Ionesco.
He's not aware of (doesn't know about) Ionesco.