Stammis means 'regular', in the sense of being a regular guest or customer at a certain establishment.
You would most often hear it used in reference to bars, restaurants or cafes, but it also makes sense if you're talking about another kind of shop or service, such as a hairdresser, bookshop, or really anything else you use regularly, and it can also be translated as 'loyal customer'.
The transfer of money is not a requirement, so you can be a stammis på biblioteket (a regular at the library) or a stammis på forumet (a regular on the online forum).
It's a noun, so you could say something like jag är stammis på baren (I'm a regular at the bar) or här är han stammis (he's a regular here), but you wouldn't say en stammis kund.
Stammis comes from the compound nouns stamkund (regular customer) or stamgäst (regular guest) but those two words sound much more formal than the colloquial stammis with the typical -is ending that pops up in a lot of Swedish slang words.
Stam itself is a noun, which can refer to the trunk of a tree and can also mean 'tribe/clan/family', so the idea is that if you're a stamkund, you're a customer who's almost part of the family.
Usually, you would say that stammis isn't a formal label; it's more about feeling a sense of attachment to the location and perhaps building up a relationship with the staff, but at some places there's a formal scheme so that you can sign up as a stammis in order to get discounts and special offers. This might be by signing up to a loyalty card (bonuskort) or paying a membership fee that is worthwhile for frequent customers.
De flesta av våra kunder är stammisar
Most of our customers are regulars
Jag känner mig som stammis här och det är så mysigt
I feel like a regular here and it's so nice