That much corrosion on the negative post could be caused by an under-charged battery, so first check that the battery is charging correctly. A battery that has been idle all night should read 12.6 volts static. If yours is closer to 12v then give it a charge for a few hours, let it have a rest and then measure the voltage again.
If the voltage stays at the charged up voltage, then check the alternator isn't on furlough. At a fast idle (15-1800rpm) the battery should report 14.3 volts. If it's less than this, or if the car's lights glow brighter when you increase the revs from idle to fast idle then the alternator is slacking and just buying a new battery, or terminal wouldn't solve the problem.
If the battery doesn't hold 12.6 volts overnight, then you should start saving up for a new battery... although I'd hang on to November/December if it's good enough for now.
To clean the terminal.. wear gloves and keep your face away from it, since it's acidic and getting a speck of acidic in your eyes is a unique experience you don't forget.. Ahhhh Ahhhh
Remove the majority of the acidic dust/crud using a baking soda, Coca-Cola, tea or even plain water solution and a small wire brush (or your room-mate's toothbrush).
Dry it all off with a paper towel.... then remove the terminal from the battery post.. being careful not to splash any residual liquid into your face. Clean the outside and particularly the inside of the terminal with emery paper (sand paper will do) or wire wool until it's shiny. Anything not metal will not transmit electricity so it should be as clean as you can get it.
The battery post can be cleaned the same way. Wire wool seems best to me.
Replace the terminal and clamp it up tight. If it's loose, you'll have resistance and then more corrosion. Then cover the whole terminal with a thin coat of battery grease, or Vaseline, or even regular wheel bearing grease. Fiat used to recommend Vaseline so the new stuff may or may not be an answer to a problem that doesn't exist.. but my old battery lasted fine with whatever grease was in the garage, so it's not desperately important.
Use "a thin coat" just to cover everything. You don't need half a tub, since it's not lubricating anything, just making the terminal air tight/waterproof since it only corrodes in the presence of moisture, regardless of what else is going on.
Inspect it every now and again (when you check the oil) and when needed give it a fresh dab of the goo.
If you do get tempted to remove the terminal and fit a new one, first get hold of a battery terminal rubber boot. These fit onto the cable and then fold over the terminal to keep dust and dirt from sticking to the greasy.. but at the moment, I'd say don't replace the terminal or the cable unless you really can't stand how the old one looks.
Ralf S.