There are two dimensions to take into account when dealing with these words. One is SIZE, the other is DISTANCE. In other words, these words differ from each other based on the size of the area that we'ra talking about and the distance of the thing we're talking about from the speaker or the listener.
SIZE is the simplest of the two: tässä, tuossa and siinä refer to a small area, täällä, tuolla and siellä to a bigger area.
So you will say for example to your friend sitting on an uncomfortable chair: "Tässä on mukava istua", referring to the soft couch you are using. When you're outside with the same friend, and the two of your are sitting in the park, you could say "Täällä on mukava istua". Then you mean that it's nice to be sitting in this park, a larger place.
Likewise, you will use tuossa to refer to a small area: "Missä on kynäni?" "Se on tuossa, pöydällä." And tuolla will be used for a bigger area: "Missä sinun mummosi asuu?" "Hän asuu tuolla, sinisessä kerrostalossa." If you'd be pointing at one specific window of the apartment building, though, you could say "Hän asuu tuossa." and your friend would know exactly in which apartment your grandma lives.
Another example will demonstrate the use of siinä and siellä. Imagine you're in the living room and your friend is in the bedroom. He yells from the other room whether you know where his watch might be. Then you can say "Etsi makuuhuoneen vaaleanpunaisesta laatikosta. Luulen, että se on siinä." The box you are referring to is a small area, so you are using siinä instead of siellä. When talking about a bigger area, you can use siellä: "Etsi rannekelloasi kylpyhuoneesta! Luulen, että se on siellä."
DISTANCE is a little more tricky, because people and things are constantly moving. A thing that is tuolla during one moment, will be täällä a little later. Tässä and täällä are used for things that are in your immediate surroundings. "Tässä on mukava istua." (on the couch) and "Täällä on mukava istua." (in the park) were used for a situation where the couch and the park were in your close surroundings. "Kynä on tuossa." and "Hän asuu tuolla." were used for things you could point at, but not touch. They were within the edge of your vision, but not touchable. "Rannekellosi on siinä." (on the table) and "Rannekellosi on siellä." (in the bathroom) were two examples where the place you were referring to was somwhere outside your line of vision. You could not see the place or item, not touch it nor point at it.
Tässä: small area, close-by
Täällä: bigger area, close-by
Tuossa: small area, close enough so you can point but not touch it
Tuolla: bigger area, close enough so you can point but not touch it
Siinä: small area, you can't see it nor touch it
Siellä: bigger area, you can't see it nor touch it
For the ones who thought this was easy: next there's a list of these words in the location cases! I suggest that you DON'T learn these yet IF the previous explanation was new to you. Practise the missä-form which was given above during a few weeks, until it becomes natural to you, and then move on to the other location cases. I could write a second post for those to remind you at that point ;)
tässä - tästä - tähän
tuossa - tuosta - tuohon
siinä - siitä - siihen
täällä - täältä - tänne
tuolla - tuolta - tuonne
siellä - sieltä - sinne