Learn French: Il mange une saucisse avec du pain. - He is eating a sausage with bread.

In French, you can almost never use a noun without an article. Because you are talking about an unspecified quantity of bread (some bread), you must use the partitive article du (which is a contraction of de + le).

Is a saucisse the same thing as a saucisson?

No, they are different! A saucisse is a fresh sausage that needs to be cooked (like a bratwurst or a hot dog). A saucisson is a dry, cured sausage (like salami) that you slice and eat cold, often as an appetizer.

The English translation says 'He is eating', so why isn't there a word for 'is' in the French sentence?

French does not have a present continuous tense like the English 'is eating.' The simple present tense Il mange translates to both 'He eats' and 'He is eating.' If you want to strongly emphasize that he is in the middle of eating right now, you could use the phrase Il est en train de manger, but usually the simple present is all you need.