fall apart means break into pieces, or stop working effectively, or become emotionally disturbed and unable to behave normally. (e.g. My new dress fell apart in the washing machine. The deal fell apart because of a lack of financing. He fell apart when they sacked him.)

fall back means retreat (e.g. The general ordered the soldiers to fall back to a safer location.)
fall back on means be able to use an alternative (e.g. It is essential to have some money in the bank to fall back on if I can’t get a job soon.)

fall down means fall to the ground or to prove unsuccessful (fail) (e.g. Many apples fell down in the storm. Do you know why our plan B falls down?)

fall in means collapse or form a line (e.g. The roof or ceiling falls in because it is damaged hurting a lot of people. The general ordered his solders to fall in.)
fall in with someone means to become friends (e.g. I thought I was falling in love with her.)
fall in with something means to accept or agree with someone else (e.g. She fell in with my idea at once.)

fall out (with some body) means stop being friends, or move out of a line (e.g. They were very good friends. But, he left home after falling out with his partner. Fall out, solders! shouted the general, they move out of a line.)

figure out means to calculate an amount, or to understand or solve something (e.g. I need to figure out how to prevent it happening again.)

fill in means write the necessary information on a form (=fill out), or to do someone's work (e.g. Please fill in the application form and posted it off. Ann will fill in for me while I’m having a vocation.)

fill up or fill something up means to make something full (e.g. I always fill the gas up when I drive. Too many spam emails are filling up my email inbox.)

find out means discover or get information (e.g. We don't know where he lives. Can you find out where he lives at this moment? You never found out who committed the murder.)

finish off means finish completely, or do the last part of something (e.g. Nearly. I'll finish it off this wine and have to go to shop to buy some more.)

fix up means arrange it or to repair, decorate or make something ready (e.g. We've fixed up a meeting with you next week sometime. They fixed up the house before they moved in.)