Browse searches every site the game's mods come from at once (for Stardew Valley, Nexus Mods, CurseForge and GitHub). A mod several sites list is one card; its badge picks the site to install from, and Preferred source (the game's settings, Mods page) sets the default. The chips beside All sources narrow to one site, and switching a site off in the game's settings (Mods page) leaves it out of Browse. A mod on the known-broken list is marked on its card, with the mod to use instead.

Open Browse. With nothing typed, each site lists its top mods.
Type a name; results update as you type.
Choose Add (Download on Nexus with Premium) on a card. It shows progress, then Added. On Nexus with a free account the button is Get, which opens the mod's files page.

GitHub: Add installs the latest release; gh auth login, when present, raises GitHub's rate limits. Nobody reviews or scans what is published on GitHub or itch.io, and the first download from either says so once.
Card details: Show › Card details picks what cards list; the Mods tab uses the same choice.
Downloads folder: an archive saved to Downloads, Mortar's download folder or a folder under Settings › Downloads › Also watch these folders is offered for install.
Extra mods folders (the game's settings): unpacked mods, or the output folder of a mod you are building. Each folder names the profile its mods install into (or the open one); they install when you add the folder and again whenever its files change, so the game loads your latest build. A change while the game runs waits for it to stop.
Hide from Browse: a card's menu has Hide from Browse and Hide everything by <author>; the game's settings list Hidden mods with Unhide. Links and imports still install a hidden mod.
Modpacks, translations and cheats: the Exclude categories picker starts with them; remove a chip to see them.
Open a mod's page: Ctrl+click a card in Browse, or middle-click a mod in the Mods tab.
Versions: Choose a version… on a card's details page installs a version other than the latest; an installed mod offers Change version…. A Support the author link shows when the mod's page gives a donation address.

Narrow and sort the results

Include categories and Exclude categories filter by category; sites without categories (GitHub, or Nexus while signed out) turn them off.
Sort orders the results. Under All sources, most downloaded and recently updated rank every site as one list, leaving out (and naming) a site that cannot sort that way, such as GitHub for downloads. Best match takes turns between sites.
Show has rows for Mods in this profile, Obsolete and Broken, each Off, Gray out or Hide (counted as "N hidden"); Obsolete and Broken can also Flag a mod with a warning mark. Broken needs a compatibility list, which Stardew Valley has.

Mods already in the profile show Added, whichever site they came from. A helper Mortar installs for a loader shows Built in.