Currencies
Star Wars: Commander Guide
Credits
Credits are the standard unit of currency for purchasing and training units. They govern pretty much all of the army-related purchases that you will make in the game, including research, and it is incredibly important to make sure you have a steady supply of these coming in. You get credits by building a Credit Market, and a Credit Vault, to store them in. Making sure your Vault is well protected is important, as that's the main way that enemies attacking your base will sap your resources. Also, having a high-level Credit Market and Vault is crucial, because you will earn Credits quicker, and be able to store more before having to spend them.If you build multiple Credit Markets and Vaults, their stats are pooled together, so your maximum Credit capacity doubles once you build a second Vault, and you will earn twice as many Credits per minute if you build a second Market. Upgrading them similarly increases the speed or capacity (as well as their hitpoints), but this only applies to the building you have selected - not all Vaults or Markets will be upgraded at the same time.
Both Credits and Alloy can also be purchased with real-world money.

Alloy
Alloy is used for building structures in your base, and upgrading certain things like Credit Vaults, Credit Markets, and defensive structures. Alongside Credits, they are the most important resource in the game.To get Alloy, you must build an Alloy Refinery. This extracts the Alloy from the ground, and the more you have, the more Alloy you will get. Much like with Credits, to be able to store the Alloy, you will need an Alloy Depot. As you might have guessed, this is roughly equivalent to the Credit Vault. The more you build and the higher level they are, the more Alloy you can keep before you have to spend it. Some upgrades will be so expensive in Alloy that you will have to save up for quite a long time, and have a high-level Depot, simply to be able to save enough of the resource.
Crystals
Crystal is the premium currency in Star Wars Commander. This means it is the way that, despite being a free a game, the developers are able to earn money from purchases of it. Usually premium currencies are associated with skipping long waiting times for things like buildings and upgrades, and indeed, in Star Wars Commander, that is also true.Probably the most important reason for someone who doesn't intend to spend real money on the game to care about Crystals is that they are required to purchase additional Droids. Droids are used to construct buildings and perform upgrades, as well as menial tasks such as clearing debris from around your base. If both of your Droids are busy (you are instructed to purchase a second one during the tutorial), then you simply cannot build or upgrade and must wait for one to finish.

You can rush the production, but that also costs Crystals. Literally, Droids have the potential to double your efficiency, and reduce the amount of time you have to spend away from the game waiting for things to be ready.