OK, so first the basic construction.
A - Savoir
"Savoir" is followed by an interrogative expression (où, pourquoi, qui, avec qui, quand, etc...):
Tu sais où il habite ? Do you know where he lives?
Savez-vous à quelle heure ils arrivent ? Do you know what time they're coming?
Sais-tu pourquoi il a dit ça ? Do you know why he said that?
"Savoir" is followed by a verb, to say "knowing how to...":
Je sais danser. I know how to dance.
And "savoir" can also be used alone.
Tu sais qu'il arrive à 3 heures? Oui, je sais.
"Savoir" can also be followed by a noun - with the idea that you have learned something by heart, memorized it. Many times you could also use "connaître par coeur" ("savoir par coeur" is not correct in French).
Je sais ma leçon = je connais ma leçon par coeur = I know my lesson by heart.
B - Connaître
"Connaître" means "to know" with the idea to be personally familiar with, to have experienced it yourself.
It is always followed by a NOUN - it cannot be followed by anything else.
In most cases, "connaître" is followed by nouns of people and places. This is where the English speaker has to be careful. When a French person says "Est-ce que vous connaissez Paris ?", in most cases he/she doesn't mean "Do you know (of) Paris", but rather "Have you personally been there?". Same with people; if you said "Je connais Brad Pitt", French speakers may think you have met him.
I have a funny story an American student told me. He was in Paris and met a lovely French woman in a café, and started to speak with her. She didn't speak any English, so he was speaking in French.
He said: "Je viens de Boston. Vous connaissez Boston ?" And the woman answered: "Non." Then he said: "Ce n'est pas loin de New-York, vous connaissez New-York?" And the woman answered: "Non." He was a bit surprised but carried on: "C'est aux Etats-Unis, vous connaissez les États-Unis?". And she answered "Non." Puzzled, he thought to himself: "Cute, but not so smart...". Of course, she understood he was asking her whether she had been there herself, not whether she knew where it was...
So, here are a couple of sentences that should clear things up:
J'en ai entendu parler - I've heard of it
J'ai entendu parler de lui / d'elle - I've heard about him / her
Est-ce que tu sais qui c'est - Do you know who it is?
Est-ce que vous savez où c'est - Do you know where it is?
Je sais qui c'est, mais je ne le/la connais pas personnellement - I know who it is, but I don't know him/her personally.
Je sais où c'est, mais je n'y suis jamais allé(e) - I know where it is, but I've never been there myself.
So typical conversations would be like:
- "Vous connaissez Paris ?"
- "Oui, j'y suis allé(e) il y a 3 ans."
- Or "non, je n'y suis jamais allé(e)."
- "Tu connais Anne ?"
- "Oui, je l'ai rencontrée chez Pierre."
- "Non, je ne la connais pas personnellement, mais je sais qui c'est. En fait, j'ai beaucoup entendu parler d'elle."
