Verbs in Russian come in two 'flavors' : perfective (eg. “пригото́вить”) and imperfective (eg. “гото́вить”).
Let's get this straight right away: most perfectives are made by attaching a prefix—and the endings of the resulting verb change in the same way they changed for the source verb.
Perfective verbs express an action, an "event" linked to a point in time. Sometimes they assert the presence of a result. You use them for sequences of actions, too.
Imperfective verbs are used for everything else: processes, states, repeated actions and for generic reference to an action (when the time of occurence is irrelevant).
In this introductory lesson we deal with the most basic pattern of use:
Being too lazy to make up many different verbs, we usually make new ones based on the old ones. The vast majority of unprefixed verbs are imperfective.
The last phenomenon is know as suppletion and only happens for a limited number of verbs and their derivatives. The English verb "to go" is another example of such behavior (its past for is "went").
Note that suffixation is very popular for secondary imperfectives. Usually only one prefixed verb is considered an "ideal match" for an imperfective verb. Others are somewhat different in meaning (or a lot different). But you need imperfective partners for these, too, so Russian uses suffixes for that:
The verb «мочь» is used to talk about the general possibility of something, and also, very often—about your ability to perform something and reach some result. Perfectives are used in the second meaning:
We do not use мочь for skills. Russian has уметь for this.
Both mean "again" and are largely interchangeable when they mean that an action from the past occurs again.
«Опять» is more popular but it's focused on staying "the same as before". «Снова» (cf. «новый») can also mean action performed "anew, from the beginning".
Only «опять» is used in «опять же» (~"besides").
When asking someone to repeat, use «ещё раз».
imperfective verbs
Perfective verbs describe events: singular, definite actions that are viewed as localized in time. They “happened” at some moment (“I made a video”, “I slept for some time and then went outside”). Or they describe a certain change of state at some "turning point" (not yet eaten→eaten, not slept enough→slept enough and ready to get up).
It is argued in a few works that "a natural" perfective is just a prefixed verb where a prefix's metaphorical meaning so conveniently overlaps the verb's own meaning, that you cannot feel any change. So don't be surprised if some vague actions have several perfective matches for a single imperfective verb.
That also means that sometimes you'd better memorize a pair even if it is technically a "poor" match. After all, in some contexts it will come in handy: