hard drive - What should I look for to have a silent external harddrive

I want a HD as a home media-server, it will always be on. Most of the time it will not be used, and it should be very silent and not to warm when not used.

Should be Regularly priced powered 2-3T external USB drive.

What should I look for to have that? Will drives usually stop spinning when not used, bringing the temperature down to a point where the fan will stop working?

Will drives usually stop spinning when not used,

Drives can stop spinning when they are not used.

How soon that happens depends on the configuration of both the drive and the OS. The OS because it can send the drive a command to spin down. Or on the drive since modern SATA drives have the capability to do this on their own.

You did mention an OS, but if you run a unix (e.g. a BSD or a Linux distribution) then look at hdparm. This is a persistent setting on the drive. (In other words: you only have to set it once).

Note that regular drive access (such as flushing log files or writing to the event log) will reset this timer. So it works best if you do not use the drive to host the OS.

This can best be combined with a small SSD (say 40GB-ish) which will not spin nor make noise. The SSD then holds the OS. The large spinning disk the data.

and the heat then so low that the fan will not be working?

The heat from drives usually is not a problem unless you run some old 15K RPM high performance drives. For a HTPC you would use a low RPM green drive. These are notably slower when doing random access (e.g. when using an OS or when running programs), but their sequential read/write speed is more than enough for movies, music etc.

As for the silent part of a drive: Get a low RPM drive. Usually those are branded as 'green'.

Not asked but: Why an external drive?

Putting the storage drive inside the HTPC would have less clutter and would be faster. (Navtive SATA access rather than using an extra SATA to USB part, an extra case and an extra power supply).

Issue: "home media server" (original question) versus "normally be connected to a router" (clarifying comment).

Sounds like you are really describing network accessible storage to be used primarily for storing movies, etc. Some home gateway routers include a USB port for that purpose but most don't.

Almost any "green" external USB drive should work well when attached to an APPROPRIATE router provided you are just using it for one TV/display at a time. Most drives are packaged as external drives without a fan at all or you can build your own by pairing a disk drive and enclosure. Metal cases usually cost a little more but dissipate heat better (simple physics provided the drive is tightly coupled thermally to the case). Make sure the enclosure has plenty of circulating air (i.e., not in a closed cabinet nor tucked behind books in a bookshelf nor in the back of a closet shelf behind winter hats and scarves) and isn't on top of/next to a heat source (i.e., not under/over a heating vent nor on top of a stereo), nor flat on the carpet under under a powerstrip and tangle of cords. For your purposes opt for a separate power supply ("brick") instead of an power supply integrated in the case. A backup fan in the enclosure may not run at all but is nice to have just in case (e.g., for an all day movie marathon or when the AC breaks).

If you want to connect the drive directly to your router/network, you probably want a NAS device instead that you can attach to the WAN port of the router. Again, almost any "green" hard disk .... (but most NAS enclosures do have fans since most are designed for commercial use with high speed often busy drives).

One additional consideration: will your network and drive provide the necessary bandwidth for your purpose. The drive or network might not support simultaneously playing movies on multiple devices.