Where can I find Alien Alloys?

There are five ways I know of to acquire Alien Alloys, ordered from most to least reliable:

• Cleaning rooms - The rooms on the lower levels of the Avenger will reward you with some units of alloy when cleaned out.
• Black market - The black market will often offer you alloys in exchange for intel.
• Map drops - From time to time "activities of interest" will appear on the global map which reward you some alloy when scanned for a few days.
• Mission Rewards - Sometimes you will receive missions to raid a supply convoy or a landed UFO. These will reward you with alloy.
• Hacking rewards - Sometimes you can get some alloy as a reward for successful hacking.

There are several ways I know of to acquire Alien Alloys.

UFO Missions - one of the possible mission types is intercepting a landed or downed UFO. Like the original game, after the mission Alien Alloys can be claimed from the craft.

Alien Machinery - There are 3 types of rooms in the avenger. One of them is "Alien Machinery", and clearing this room type will give a small amount of Alien Alloys.

Supply Raid Missions - Supply Raid missions (Advent Trains) can result in Alien Alloys.

Black Market - The Black market will occasionally offer Alien Alloys. You must buy these lump sump for intel, and they aren't always available.

See also Philipp's answer, above.