Astels are the game's headline feature, and almost in the name of the game itself, so I thought I'd explain what they do.
Basically, astels are the game's pet system. In a sense, all classes are pet classes. You can play a tank with pets, a healer with pets, or a damage dealer with pets. By lore, astels benefit from coming with you, as they need to have a master just to survive. In some cases, you acquire an astel when you save it from imminently dying. You'll want at least one astel out most of the time, and sometimes more.
In addition to your health and mana bars, you have a third bar for your "AP". I'm not sure if AP stands for Astel Points or Atra Power or what, but it's basically your ability to summon and keep astels out. Your AP bar maxes out at 900 AP, and as best as I can tell, that maximum can't be changed. It normally refills at a rate of 30 AP per second. There are a handful of fairly restricted ways to increase that a little, but for the most part, 30 AP/sec is what you get.
There are three different types of astels: saviors, guardians, and servants. The game currently has 2 saviors, 6 guardians, and 26 servants.
Servant astels are naturally the weakest. They cost 100 AP to summon, as well as 20 AP/sec to maintain. Thus, if you only use servant astels, you could have two of them out a little less than half of the time. Servant astels generally look young, while the other types look much older, so for the most part, you can tell them apart visually.
Guardians are the next type of astel, and naturally stronger than servant astels. They cost 200 AP to summon, as well as 25 AP/sec to maintain. Thus, while you can have a second astel out occasionally in addition to a guardian, your recharge rate will be slow with just a guardian out.
The final type is savior astels. Savior astels are naturally very strong. They cost 750 AP to summon, but no AP to maintain. However, they can only be summoned for 45 seconds before they are gone.
You can manually summon or dismiss astels whenever you like for the most part. If you run out of AP, the game will automatically dismiss astels as necessary to make your AP recharge rate positive. It starts by dismissing the last one summoned.
When you unsummon an astel, it cannot be summoned again for a period of time that depends on how much HP the astel was missing. Thus, if you dismiss an astel with full health, you can resummon it a few seconds later. If an astel dies, or if you die while the astel is summoned, that counts as the astel being dismissed with zero health. When a savior astel's time runs out, that also counts as it being dismissed with zero health.
You have an astel bar with room for eight astels on it. Only astels on that bar can be summoned. You can freely swap astels on or off of the bar whenever you like. I haven't tested whether you can do that while in combat (which you probably shouldn't because you're in combat), but you can readily swap astels on your bar between battles without any limit.
There are three main ways to power up your astels: astel level, astel star rank, and star gems. Astel star rank is based on how many star points you have earned for that particular astel. All star points are earned in multiples of 10. It takes 10 star points (basically, anything that isn't zero) to reach rank one. Otherwise, it takes 10 * (2(n-1))^2 SP to reach rank n. The maximum star rank is 7.
When you acquire an astel, it generally comes with a given star rank. This is really the number of SP that it gives you, which will be added to what you already had if you previously had the astel. For example, getting a one star astel gets you 10 SP, two star gets you 40 SP, three star is 160 SP, and four star is 360 SP. If you get the same one star astel repeatedly, it adds 10 SP for that astel each time, so that it will become a two star astel the fourth time, a three star the sixteenth time, and so forth.
There are several ways to acquire astels. Some dungeon bosses will drop a random astel card, worth 10 SP for some astel. All dungeon end bosses do this except at scenario difficulty, and many dungeons also have a mid boss that drops such a card. Once you reach level 50, you can do dailies to get a card with a one to three star astel each day. For both dailies and dungeons, you can pick one particular pool of servant astels to drop from, so that groups of five to seven servant astels are semi-farmable by repeatedly choosing the same pool. You can't choose to farm a particular astel. These also have a small chance of dropping a random guardian or savior astel. Because the servant astels are semi-farmable and the other types are not, it's easier to get servant astels to higher star ranks.
Certain main storyline or "important" quests will grant a particular astel as a reward. These start out as one star, but soon start giving two star astels, and eventually three star. Several particular astels have three or four star versions awarded for Star's Tale progress. These are predictable, but one-time rewards. You can get Rota, Corvus, and Aquarius to four stars by getting 100% Star's Tale completion in particular zones, but no other astels can go above three stars without relying on dungeons, dailies, or the card rewards from certain achievements.
Astels have experience levels just like players do. Whenever you get experience, all astels on your bar get exactly the same amount of experience that you do, with only the exception that astels don't get the 10% bonus experience for certain events. Both mob killing and quest turn-in experience is granted to astels, in addition to you. Only astels on your astel bar get experience, but they get that experience regardless of whether they are summoned.
Astel stats increase greatly as their level does, just like players. In particular, their max HP increases. While generally a good thing, this means that a high level astel could have a lot of HP missing when you unsummon it, which increases the wait to be able to summon that particular astel again. For most astels, a trivial wait if the astel dies early on eventually grows to several minutes at the level cap. For savior astels, it can eventually approach half an hour.