1) There are two different spellings: asseoir (traditional, much more common, the one I prefer and the one I use for this answer), and assoir (1990 modernized grammar, typical of some official texts, but still less common).
2) There are two strictly equivalent, fully correct conjugation forms to asseoir (to make or help someone / something sit down) or s'asseoir (to sit down, much more common). You can use both of them, independently of the chosen (old or modernized) spelling which only applies to the infinitive.
The first, and less usual form is (for s'asseoir):
- Je m'assois
- Tu t'assois
- Il/elle s'assoit
- Nous nous assoyons
- Vous vous assoyez
- Ils/elles s'assoient
It is similar to other stem-changing verbs like employer or nettoyer (1st conjugation group), so this conjugation is the one of a 1st group verb, except for the infinitive (that's why it belongs to the 3rd group).
This form is very common when used with the first, second or third person singular, and the third person plural probably because of the similarity to the infinitive. However, the first and second plural forms are uncommon. I can't give any scientific reason for this, but it sounds like Old French, and looks like a very common grammatical mistake made by little children for 3rd group verbs.
The second form, considered more common, and by some more polite, would be:
- Je m'assieds
- Tu t'assieds
- Il/elle s'assied
- Nous nous asseyons
- Vous vous asseyez
- Ils/elles s'asseyent
This is a "pure" 3rd group irregular conjugation, more here
3) Practical examples, meanings, and Common mistakes:
- My old middle school teacher, telling his class to sit down: "Assoyez vous!" (Could someone give me a proper, context adapted English translation?)
- What I would recommend: "Quand je m'assieds dans un bar, j'aime être servi rapidement" or "Je m'asseois toujours à cette table, c'est ma préférée".
- Most French people will use either one of the two forms (there could be regional differences), or, when using the first one, will almost certainly switch to the second form to avoid using
nous nous assoyons or vous vous assoyez . Example: "Je m'assois sur la chaise. Si vous vous asseyez sur le canapé, ne mettez pas vos pieds dessus!"
- Je m'assis / Tu t'assis / Il s'assit is often mistaken (especially among teenagers) for the present form Je m'assieds / m'assois. However, this is the "Past Simple" (I sat down). It is a common mistake among French / French-speaking people, but it is an ugly mistake.
Meanings, similar verbs :
- Asseoir: to make someone sit down, or to help someone "stand his ground" in an argument. Il a assis sa position avec des arguments solides / Avec l'apparition de l'iPhone 2000, la companie "Pomme" a assis sa position sur le marché du téléphone portable or J'ai assis le bébé sur la chaise pour le repas.
- s'asseoir sur quelque chose: to (intentionally) ignore or put away something (usually an argument, an opinion, a law): Il s'est assis sur le règlement et a fait ce qu'il voulait. You can imagine the guy in my example putting the rules under his bottom. This is not familiar, but using this construction means you clearly insist on the intention not to take the object (here the rules) into account.