The verb to hack basically means to cut something in a rough way, with a lot of energy or many times. It can also mean to cough loudly.
There are also several examples of hack as a phrasal verb. If something is hacked about (normally used in the passive), a lot of changes are made to it in a rough or careless way, as in ‘The old door had been hacked about over the years’.
If you hack away, you cut something several times, again in a rough or careless manner, as in ‘He grabbed a branch and started hacking away’ or ‘She hacked away at the bushes with her knife’.
The verb to hack down means to cut down, but if you use it, you are probably expressing disapproval, e.g. ‘Rainforests are being hacked down with no thought for the environment’.
Hacking into something, on the other hand, has a quite different meaning. If you hack into someone’s computer, you connect to it secretly (and illegally) in order to obtain confidential information, often for financial gain, as in ‘Criminals try to hack into the computer systems of banks and transfer large amounts of money’.
Hack off has the literal meaning of cut off, as in ‘She hacked a piece off the block of cheese’, but it can also mean to annoy someone, as in ‘These constant delays are really hacking me off’ or ‘She’s really hacked off about the changes they’ve made to her timetable’.
Finally, if you hack something up, you cut it into small pieces, as in ‘The killer hacked up the body’, or, with the meaning of cough, you try to remove something from your throat or lungs in a loud, rough way, as in ‘It sounded as if he was hacking his lungs up’.