The usual basics are probably still needed. Food, tools and clothing/adornings (I'll assume they don't need clothes to protect them from the element (singular because it's just water) which they should be adapted to, but likely from the point of modesty or fashion).
Grooming services will be needed and basically any activities which comes with the level of technological advancement you desire. A hunter/gatherer tribe does not need much in terms of economy as there is very little degree of specialization.
But as soon as you need to specialize you get the need for services such as childcare and schooling. With medical advancement you need to set aside tribe members for various medical professions. With religion you get a priestly caste and probably also a need for a military/constabulary profession (to keep the flock in line and to enforce the laws of Neptun/Dagon/Old Father Shark). These are no longer producing their own food and as such need to sell their service (collect tax).
And of course you have the old trope about the "oldest profession" though if they procreate like fish that might be moot. (
Also you probably need to invent shame and sexual repression for it to be a viable trade, so presumably priests come first. Properly refuted by Perkins, we agree that shame and sexual repression does improve the market though.)
Also, the question (to me at least) is what kind of trade will occur between land and sea. If they are isolated and have no contact with humans, just pick a technological level (pre-civilzation, ancient, medieval, pre-industrial, post-industrial, information age... whatever suits you story) and assume that there will be analogues to whatever land economy you're mirroring. (Herding dolphins instead of cows, harvesting kelp instead of wheat, using lava flows and heat vents to cook food and make tools.) If they have contact with the other element you want to think about what's made where. Perhaps tools are made on land and traded for pearls and corals (or hard to fish from the surface fish and crustaceans).
G0BLiN asked, so I'll see what I can think of. Paper currency is of course out and metal might be unfeasible as well depending on the tech level. (Whether they can use lava flows to forge or not.)
But there are a lot of other viable currencies.
- Commodity money. The Mayans used cocoa beans. The world of Metro 2033 uses pre-collapse ammunition. It's basically assigning a base unit for barter trade. I'm not enough of an economist to say how this affects inflation. Will a bumper crop ruin the economy or will supply and demand automatically adjust price levels? This could be anything (driftwood, tubers, shark eggs) but preferably small enough to carry around in a pouch around your neck for when you go shopping.
- Shell money. Pieces of coral, pearls polished rocks, sea shells. Anything shiny that sentients like. There are a great number of precedents here.
- Stone money. This is my favorite, seeing as how it kind of works like our current system. A value is given to a thing, usually a large rock that has been worked in some way. The Rai stone is a large carved stone disc (or wheel) that can weigh up to 4 metric tons (8,800 lb). This is rarely ever moved but ownership is agreed upon by everyone and using it for trade is just transfering the ownership. It is a system which depends a lot on honesty (especially in lieu of solid record keeping) but from a cynical perspective it does not differ that much from having money in the form of a number in a database field somewhere.
- Domestic animals This is pretty much a form of commodity money but based on livestock. If your merfolk herds some sort of sea animal they can be used as the base for an barter economy (Fish might be tricky, but crustaceans or cephalopods may work). Camels have been known to be used like this on land.