This is one of the most difficult topics to learn for most people studying Spanish. Although Por and Para both mean "for" in English, they have a different meaning in Spanish according to their position in the sentence and should not be used indiscriminately. Which of them is to be used in a sentence can only be determined according to its meaning. Por has more uses than para.

Spanish dictionaries translate por as follows:
by, for, per, to, through, at, about, as, because of, via, along, around

and para as:
for, to, by, for the purpose of, in order to, so as to

The uses of Por:

1: To thank or ask for forgiveness
Thanks for the help - Gracias por la ayuda

2: Duration or time span, in the sense of "during"
I waited for 3 hours - Esperé por tres horas

3: A density, share or frequency, mostly described by means of "per"
Three times per/a day - tres veces por día

4: Exchange, in the sense of "for" or "in place of"
I give him this one in exchange for the other one - Le doy ese por el otro

5: Support, fondness - "for"
I am for the white team - Yo apuesto por el equipo blanco

6: Indication of a specific time span
He comes for the day - Viene por el día

7: Transport and communication - also replaces "via" or "by "
He travels by bus - Viaja por autobús
He is talking on the mobile phone - Está hablando por teléfono móvil

8: Grounds and causes - due to, because
He doesn't hear because he is deaf - No oye por ser sordo.

9: Multiplication
Three multiplied by two makes six - dos por tres son seis

10: Through, beside, along or around a physical location, either with or without motion (not a specific destination)
I drive through the tunnel - Conduzco por el túnel
There are a lot of camping places beside the lake - Por el lago hay muchos sitios de camping.

11: In the passive mode, when it replaces the word "by"
The damage was caused by Mike - El daño fue causado por Mike

12: As a positive initiative: "Estar por" means "to feel like" or "to want to do something" or "to be about to do something".
Estoy por llamarla - I am about to call her

The uses of Para:

1: A specific physical destination or in direction of a specific location (up, down, skywards)
The train to Chicago has left - El tren para Chicago salió;

2: Purpose of an object or an action. Used in English in situations such as "to do something with it", "serves for". In Spanish it is mostly used with the verb "ser".
The chair is to sit on - La silla es para sentarse

3: In front of a receiver
This is for you - Esto es para ti

4: In comparison to something else, and contrary to general expectations. "Other than expected" could be added to the sentences. Replaces the word "for".
Well done for a beginner! - ¡Está muy bien para un novato!

Apart from these rules, "Por" is used in many expressions:
Por favor (please), por allí (over there), por cierto (in fact), por eso (therefore), por fin (at last), por ningún lado (nowhere), por ejemplo (for example)