Paxa means 'to reserve', but only in certain contexts.
If you're talking about reserving a hotel room, a table at a restaurant, or a seat on a train, you'd use the verb reservera — simple enough to remember, since it's close to the English. You can think of these as 'formal' reservations: you'll often get a written confirmation. Another alternative is boka (to book), as in kan jag boka ett bord? (Can I book a table).
By contrast, paxa is used to describe 'informal' reservations. For example, if you put your coat or bag on a chair at a cafe to show that's 'your' seat, or if you call out a preference for a seat in a car while the whole family is walking towards it.
English equivalents, depending on which national and regional dialect you speak, could be 'to call shotgun' (although this is very specific, referring to sitting in the front passenger seat of a car) 'to call dibs' or 'to bagsy'. Swedish uses paxa to cover most of the same contexts.
So, how do you paxa?
Simply shout ‘pax för X’ and all decent people will understand that the spot or it is yours. You've paxed it. Note that this is an informal, colloquial term, so you wouldn't want to walk into a work meeting and shout pax för stolen vid fönstret! (I call dibs on the chair by the window). But if your office has a relaxed working culture, you may well hear phrases such as jag har paxat hans kontor! (I've called dibs on his office).
Children often use it while playing: pax för cykeln! (I call dibs on the bike!), pax för framsätet! (I call shotgun!). In other games, they'll also use it to choose which character they will play: pax för att vara Pippi (dibs on being Pippi). You can use pax with the verbs få or ha, but can also use the verb paxa on its own, for example jag fick pax på cykeln or jag har paxat cykeln (I called dibs on the bike).
Another context when you will hear pax used among grownups is in reference to seats or tables at a cafe, bar or restaurant. If you put your coat or bag down on a seat while you queue for food, to save yourself that spot, you've paxat that seat. Whether or not this is bad etiquette is a bit of a contentious question, but you should definitely avoid it in popular eateries, where you'll sometimes see signs warning paxa ej platser! (no saving seats!).
It's not entirely clear where this word comes from (if you have a theory, please let us know!), but in Swedish pax and pass are both used to describe a 'safe' space in children's games, where a player can't be caught. The English equivalent might be 'den', 'base', or 'dell', but it varies a lot depending on the region. in Swedish you could say detta området är pass/pax (this area is safe). This probably comes from the Latin word pax meaning 'peace', and it's likely that the phrase pax för in the sense of 'dibs' and the verb paxa come from the meaning of 'safe space'.
Men jag hade ju paxat!
But I called dibs!
Den platsen är paxad
This seat is saved