Depending on who is doing the talking, Icelandic verbs can be divided up into 5, 6, or even 7 (I think?) groups for their present tense formation. As my most reliable grammar references these days is Íslenska fyrir útlendinga*, I shall be grouping verbs as they do. Note, however, that the groupings are all shuffled around when we talk about the past tense, where verbs are divided into weak and strong.
The easiest groups to conjugate are groups 1-4, and they also contain the vast, vast majority of verbs, so we'll start there. First, recall (or learn, if this is your first time 'round) that Icelandic infinitives usually end in -a (e.g. tala - to speak) or -á (e.g. sjá - to see). Some verbs also end in -ja, where the j is also sometimes treated as part of the ending. (A rather common exception to this is the verb þvo - to wash, which we will not be dealing with today!) These groups are distinguished by how they form their 1st person singular, so if you remember the 1st person singular of these verbs, you know how to conjugate the entire present tense (and, as we will reveal later, this gives you some hints about the past tense).
We'll start conjugating the singular forms. This may seem weird, but there is a method to my madness.
Group 1 verbs are those that end in -a where this is also a part of the stem of the verb. This means that, unlike other groups, the -a is not simply an ending used for the infinitive; it is carried through in many conjugated forms. Such verbs can be identified by the fact that their 1st person singular form is the same as the infinitive, for example the verb tala - to speak becomes ég tala - I speak. In the 2nd and 3rd person, an -r is added to the end:
Group 2 verbs drop the -a in their infinitive add an -i to the stem to form the 1st person singular, (e.g. heyra - to hear, ég heyri - I hear). These verbs get -ir in the 2nd and 3rd person endings:
Not too bad, right? Next, we come to the slightly difficult groups, 3 and 4 (don't worry, you can still handle them). They form their verbs similarly, and many people treat them as the same group (I probably would in the absence of my trusty grammar bible). An example of a group 3 verb is telja - to count. This is one of those aformentioned -ja verbs, and the entire -ja disappears in the singular conjugations. In fact, many of the -ja verbs belong to this group. If you see a -ja verb you don't know, this would be the group to guess (though this is by no means a hard and fast rule, byrja - to begin is a group 1 verb, for example). (This is not to say that all verbs in this group end in -ja, as there are plenty that do not and simply drop their -a in conjugation). The 1st person singular is simply the stem of the verb with no ending, so telja becomes ég tel - I count. The 2nd and 3rd parts get the ending -ur
Group 4 verbs are formed with exactly the same endings. The only differences is that the vowels of their stems change according to the i-shift, called i-hljóðvarp or B-víxl in Icelandic. What is the i-shift, you may ask? The i-shift is one of two vowel changes that permeate Icelandic conjugation and declension. Its details will be the subject of an upcoming post. For now, it suffices that, as you must learn the 1st person singular along with a verb, note if there is a vowel change. As an example, take the ubquitious verb taka, to take. It becomes ég tek - I take. The vowel shift carries through to the 2nd and 3rd person singular, but the endings are the same as group 3
Note that the split between group 3 and 4 is particularly artificial. They have the exact same conjugation patterns, but group 3 verbs are those that just happen to have a stem vowel that is unaffected by the i-shift, whereas that is ALL that group 4 contains. Therefore, you will never encounter a group 3/4 verb that could take the i-shift but doesn't, and can happily treat them as one group, if you wish.
If the situation seems a bit complicated for conjugating the singular forms, take heart! The plural forms are the same for all verbs in the present tense, including the more troublesome verbs that we have no yet discussed. Group 1 verbs drop the final -a of their stem, and then all verbs get -um in the 1st person, -ið in the 2nd person, and -a in the 3rd person (that's right, the 3rd person plural is always the same as the infinitive).
Okay, so you might have noticed that there's some strange stuff going on.
First, if you aren't familiar with the u-shift, tölum probably looks very strange to you. Remember how I said that the i-shift was one of TWO vowel shifts? Well, the u-shift is the other one (u-hljóðvarp or A-víxl in Icelandic). It is much simpler, because it only affects one vowel: a. If an ending starts with a u (as -um does), and the preceding vowel starts with an a, it becomes ö if it is the stressed syllable, or u if it is unstressed. And why tökum? The i-shift only affects the singular conjugation of taka. So in the 1st person plural, it would be *takum, but because of the u-shift, it becomes tökum. This can also affect a chain of a's. For example, the past tense of tala is ég talaði, but við töluðum. The first a, which is stressed, becomes an ö, but the second a becomes u. Do not stress about the formation of the verb, the important thing to get is the vowel shift. Note that this only affects an a that is in a syllable next to the ending that starts with u or another u-shifted a. The presence of any other vowel will break the chain. Want an example of this? I can't bloody find or think of one, so if someone has one, please share =p The u-shift applies not only to verb conjugation but declension as well, and I will cover it more fully in a separate post.
The second thing that you might have noticed is that the j came back for some verbs ending in a. It does that in the plural. The only exception is that you never insert a j between g or k and i (e.g. syngja (to sing) - við syngjum, but þið syngið). The reason for this delves deeper into phonetics than you want to go after you've just learned all this about Icelandic verbs.
That's all for now. I will cover the remaining groups of verbs in another post. For now, get conjugating, ask questions in the comments, etc.
*The example verbs for groups 2, and 3 are taken from Part II, Chapter 1 of Íslenska fyrir útlendinga.