Where can I find Alien Alloys?

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.

You can also acquire alien alloys as a hack reward (albeit, with a low probability, dependent on your squad member's tech score) upon a successful hack, seen in the screenshot below:

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.