The verb makes it different:
with an action verb (manger, boire, prendre, attraper...) you have to use the partitive: du fromage (= some cheese)
with an appreciation verb (aimer, adorer, préférer, détester, apprécier...) you have to use the definite article le, la, les: j'aime le fromage (generality = all types of cheese)
As Sitesurf points out, the nature of the verb changes possible translations of a sentence.
With an action verb like manger, you can't eat all the cheese in the world. You can only eat some of it, or alternatively, a particular piece of cheese. The English, he eats cheese, clearly does not mean he eats only a particular piece of cheese so it should not be translated that he is. Il mange du fromage means he is eating only some of it. Not all cheese, not a particular piece of cheese but just some of it.
An appreciation verb is the opposite. The English, I like music is taken to mean the speaker likes all music. If he didn't like all music he would have said I like some music or I like the music.
In both examples (I like cheese, I eat cheese) English speakers just drop the article and let the listener figure out what was meant. Of course in French, you can very seldom just drop the article for convenience of speaking. To provide an article to express generality they have given le/ le/ les a dual role.
le/ la/ les = the - that/ those one/s right there - specific
le/ la/ les = the - all examples of something, the idea of something - general
du/ de la/ des = of the - usually taken to mean some
So...J'aime le fromage = I like all examples of cheese or I like that particular cheese.. depending on context.
...Je mange du fromage = I am eating some cheese.. because I can't be eating all of the cheese in the world.
To be precise "du" doesn't really mean "some", even if you can often translate it this way. "Du" is just a contraction (just as "doesn't" stands for "does not"), it is the sort form of "de le" which is incorrect. There are verbs requiring "de" as preposition (as in English you have to use "from" or "of" with some verbs)
Because the subject is "nous". French has extended conjugations with endings changing with almost every grammatical person:
je mange, tu manges, il/elle/on mange, nous mangeons, vous mangez, ils/elles mangent.
Regarding "mangeons" in particular, an -e- is inserted between the 'g' and the -ons ending, so as to keep the soft G pronunciation.