Twitch (soon to be Overwolf) -- launcher arranged around the concept of click-n-play modpacks, though you can still do so there's not really any need to tinker around with individual mods
GDlauncher -- basically Twitch but without the bugs
MultiMC -- a launcher designed (at least initially) with custom-modpack creation in mind. While the newest (development) versions have a modpack browser, it's not quite as modpack-level as the above two clients. That said, everything is very easily accessed so it's pretty easy to add/remove/update mods
The above are essentially the gold standard, the toptier goto's that everyone recommends. There are also other clients that you might like:
FTB Launcher -- this is a curated launcher moreorless designed to feature FTB packs. The old version was abandoned somewhere around Minecraft 1.7.10, but it was recently updated when FTB and Curse ended their relationship earlier this year. The plan was to host FTB and approved 3rd-party packs, but it sounds like perhaps nobody really wants to work with them.
AT Launcher -- I think this is another curated launcher, it's pretty lightweight-feeling. I don't recall it having many packs, but I found it decent.
Technic Launcher -- never used it, sounds like it's pretty old/outdated. I hear it's hard to add/update packs.
MagicLauncher -- never used it, no idea on if it's still supported.
The above launchers are all designed to make modded Minecraft easy and convenient to get into. The vanilla launcher is fully capable of the same stuff, but it's tedious to accomplish.