The name for the difference is "strong" versus "weak" verbs. Strong verbs have a vowel change in the past, weak verbs add -ed. (Life gets slightly more complicated: for instance, there are irregular weak verbs, but that's not relevant to this question.)
It basically has to do with how frequently the words are used (or were used in the past). We learn to use these other past forms by hearing them repeated. If we don't hear one enough, the strong past form (the one not formed with -ed) sounds "wrong" to us, and we use the -ed form.
You can see this with young children of the right age, who routinely try to use -ed on strong verbs. When a word is used rarely enough, people grow up without ever getting comfortable using the strong past form, and continue to use -ed. They then pass that on to their children, and so on.
So over time, strong verbs become weak if they're used infrequently. You can see this dynamic in play today with words that are on the edge of the necessary frequency; for instance, what is the past of "dive": "dived" or "dove"? Quite possibly in a generation or two, "dove" will sound archaic and "dived" will be standard.
Do verbs have explicitly "plural" versus "singular" conjugations, or do they simply have the 6 conjugations (per tense), and "to be" just happens to look like "plural" versus "singular" by mere coincidence?
I'd say that you are right. English verbs have, in principle, six forms, though the maximal number of distinct forms is four, and that only for the present tense of to be. The verb form is entirely determined by the subject. The fact that the subject pronoun you can be both singular or plural doesn't change this at all: you always requires the same verb agreement.
BTW, you can demonstrate that the English "singular" conjugation is in fact not a unitary conjugation by throwing back in the old 2sg thou:
I am, thou art, he is
I was, thou wast, he was
I say, thou sayest, he says
The 1sg and the 3sg are only the same in a few verbs and tenses, and the old 2sg is always different from both of them. The plural forms of we/you/they are always the same for all verbs in all tenses, so as far as that goes it's reasonable to talk about a "plural" verb form. But there is no one "singular" verb form.
Best Answer
It may be that your teacher was wrong, or it may be that one of you confused "regular" verbs with "strong" verbs.
In English linguistics (and Germanic languages in general), there are two classes of verbs known as the "strong verbs" and the "weak verbs". The strong verbs are those whose past tense and past participle are formed by ablaut, which is a vowel change in the stem of the verb. Strong verbs usually have a past participle which is different from the simple past form, and may have their past participle formed by -en.
The "weak verbs" are those whose past tense is formed with -d, -ed, or -t. These rarely or never have a past participle that's distinct from the simple past form.
In principle, this distinction is orthogonal to the regular/irregular distinction. As the previous examples show, there are both regular weak verbs (lift/lifted) and irregular weak verbs (say/said). In older forms of English there were both regular and irregular strong verbs as well, but as time has gone on the "regular" strong verb patterns have applied to fewer and fewer verbs.
As a result, in modern English all strong verbs are irregular, and all regular verbs are weak verbs. However, in Old English the strong verbs were more common than the weak verbs, and most of the strong verbs were considered regular verbs. This confusion between "regular" and "strong" verbs is probably at the root of what your teacher said.