Just when you thought you'd nailed it with Polish verbs, conjugating correctly for each subject, using the right stem alterations for past tense formations in every gender, and expanding your verb vocabulary quickly, you get this: perfective and imperfective aspects.
So, what are they? Well, actually, they're pretty much the same as English tenses, and remember English has far more verb tenses than Polish could possibly imagine. But, instead of calling a particular verb form the 'past perfect' or 'future continuous', like we do in English, Polish distinguishes between what it calls 'aspects': The perfective and the imperfective. This means you don't have to deal with any of that cumbersome and tongue-twisting tense jargon like in English; hopefully you'll learn to see how logical the Polish way of doing this is.
In the English past tense we can say both 'I did' (known as 'past simple' tense) and 'I was doing' (known as the 'past continuous' tense, because it expresses an action in the past that is still going on), and while there is actually no direct equivalent for the Polish perfective and imperfective aspects (mainly because when we apply this past tense logic to future tense situations it's difficult to maintain them), these come pretty close to showing how the Polish system works in English (They are listed here respectively).
So, knowing when to use the right past tense verb, perfective or imperfective, will affect the meaning of your sentence. The central point here is time; whereas one will indicate a finite and complete past action (perfective), the other indicates an on-going, unfinished or untimed action in the past (imperfective).
Many perfective verbs are formed by adding a prefix to the beginning of the word. For example, and here I'll use the masculine personal past tense form, if I wanted to say 'I was doing', I would say robiłem, which is the imperfective verb aspect of the infinitive robić. However, to say 'I did', that is to indicate the action has been completed in the past, I would use the perfective zrobiłem.
Perfective prefixes vary from verb to verb, but as they always use the infinitive verb form and simply add to it, it's quite easy to identify which is which. Common prefixes are z- (as with zrobiłem ), ze- (‘zerwałem’, 'I ripped off'), po- ('poszedłem', 'I went'), u- ('ugotowałem', 'I cooked' ), za- ('zapłaciłem', 'I payed') and na- ('napisałem', 'I wrote').
There are also, of course, irregular perfective verb forms, where it's difficult to see what imperfective verb is being mirrored perfectively. For example, the imperfective verb meaning ‘to have been watching', oglądałem, is nothing like the perfective verb 'to have watched', obejrzałem, and 'to have been speaking' mówiłem, not at all like powiedziałem ('I said'). There aren't too many of these but they are worth learning separately as exceptions to the rule.
Often, when a verb may not be perceived to have a perfective form in terms of meaning, its prefixed form is still used to convey something else. It's also common to use a stem alteration, along with a different imperfective verb, to produce verbs with new and related meanings in the perfective form. The imperfective form of these is then generated in turn by removing any associated prefixes on the new stems. So, the perfective pomogłem (I helped), once formed from the imperfective, meaning-related móc (to be able to), has the imperfective infinitive of pomagałem (I was helping).