The details.
What actually is hosting?
Your website is a set of files. Hosting is the rented computer that stores those files and serves them to visitors around the clock. The domain is the address; hosting is the building.
When hosting stops, the site stops. That is the whole relationship.
Do I need to manage it myself?
No. On a modern small business site, hosting runs in the background and can go long stretches with no attention at all.
What you need is not management skill but ownership: the account in your name, with the renewal notices coming to you.
Whose name should be on the account?
Yours, wherever possible. If someone else holds the hosting and the relationship ends, getting your site back can turn into a negotiation.
It is fine for a designer to do the work inside the account. It is not fine for them to be the only one who can open the door.
What does hosting cost?
For a small business site, the going range is a few dollars a month to a few tens of dollars a month, depending on what the site does. A simple brochure site sits at the low end.
Be suspicious of two extremes: free hosting that makes its money from you in other ways, and a big monthly fee for a site that needs very little.
What should I ask before signing up?
Whose name is on the account, what happens to the site if you leave, who gets the renewal notices, and what the renewal price is after any first-year deal.
The answers should be boring. Surprises in hosting are rarely the fun kind.