We already learned the most common type of negative verbs; the ones that end in 「ない」. However, there are couple more different types of negatives verbs. The ones you will find most useful are the first two, which expresses an action that was done without having done another action. The others are fairly obscure or useful only for very casual expressions. However, you will run into them if you learn Japanese for a fair amount of time.
Way back when, we learned how to express a sequence of actions and this worked fine for both positive and negative verbs. For instance, the sentence "I didn't eat, and then I went to sleep" would become 「食べなくて寝た。」 However, this sentence sounds a bit strange because eating doesn't have much to do with sleeping. What we probably really want to say is that we went to sleep without eating. To express this, we need to use a more generalized form of the negative request we covered at the very end of the giving and receiving lesson. In other words, instead of substituting the last 「い」 with 「くて」, we need only append 「で」 instead.
Hopefully not too difficult. Another way to express the exact same thing is to replace the last 「ない」 part with 「ず」. However, the two exception verbs 「する」 and 「くる」 become 「せず」 and 「こず」 respectively. It is also common to see this grammar combined with the target 「に」 particle. This version is more formal than 「ないで」 and is not used as much in regular conversations.
Finally, we cover another type of negative that is used mostly by older men. Since 「ない」 is so long and difficult to say (sarcasm), you can shorten it to just 「ん」. However, you can't directly modify other words in this form; in other words, you can't make it a modifying subordinate clause. In the same manner as before, 「する」 becomes 「せん」 and 「くる」 becomes 「こん」 though I've never heard or seen 「こん」 actually being used. If you have ever heard ｢すまん」 and wondered what that meant, it's actually an example of this grammar. Notice that 「すみません」 is actually in polite negative form. Well, the plain form would be 「すまない」, right? That further transforms to just 「すまん」. The word brings up an image of おじさん but that may be just me. Anyway, it's a male expression.
You can even use this slang for past tense verbs by adding 「かった」.
There is yet another version of the negative verb conjugation and it uses 「ぬ」 instead of the 「ない」 that attaches to the end of the verb. While this version of the negative conjugation is old-fashioned and part of classical Japanese, you will still encounter it occasionally. In fact, I just saw this conjugation on a sign at the train station today, so it's not too uncommon.
For any verb, you can replace 「ない」 with 「ぬ」 to get to an old-fashion sounding version of the negative. Similar to the last section, 「する」 becomes 「せぬ」 and 「くる」 becomes 「こぬ」. You may hear this grammar being used from older people or your friends if they want to bring back ye olde days.