Alright, there we go. Felt good, right? You can see it just in how the Sprite's Icon looks, yeah? It's no longer slowly returning to Resonance. You'll have this one until you run out of tasks. Just keep in mind that you have a limited number of registry slots in your Living Persona; one for each point of Charisma you have. You can try to register with all the slots filled out, but the Sprite will not let you forge the initial connection.
Now that your Sprite is registered, it's capabilities can go much further than just existing for more than 8 hours. First bonus to logging your favorite Sprites; Remote Tasks. Usually, telling a Sprite to conduct a task in a node you aren't directly connected to causes it to immediately return to Resonance once the task is completed, even if it could have done more tasks. A registered Sprite, however, will just go on standby after completing the task. Keep in mind two things, though; you can still only give it the one task to do, and it will remain idle on the node after it is done. You can give it further instruction to return to you or a safe network once it's done, but you will need to give that instruction beforehand unless you go into the node later.
If you want to give an allied decker (or anyone) help without being nearby, you can loan out your registered sprites to them! Of course, the Sprite helping an ally doesn't make your ally Resonant; they can't learn from a sprite, thread or sustain a complex form, but anything else can be requested. Imagine this being like giving someone else an account with user access. You are still the admin, and if you overwrite or overrule a command, the Sprite will listen to you over the other person.
In situations where the Sprite itself may not be very useful, for example, a Courier in a cybercombat scenario, the Sprite can give increase the power of one of its Technomancer's complex forms. The increase and the time are based on the Sprite's rating; a Rating 3 Sprite can increase the Technomancer's Attack form by 3 for 3 Combat Turns.
Similarly, a Sprite can take on the -2 dice pool modifier a technomancer gains by threading a complex form. Just like with increasing the power of a complex form, the amount of time a Sprite can do this with a single task is equal to the Sprite's rating. Keep in mind that if the sprite is destroyed, by decompile or outright cybercombat, the entire sustained form will end, forcing you to rethread if it's still needed.
When attempting to learn a new Complex Form, a registered Sprite can add their rating to the Extended Learning Test, assisting the Technomancer in learning. Only for Complex Forms, though. There are no sprites who can help with learning other skills. Not now, anyways.
If your sprites run low on tasks, you can re-register them! The process is the exact same as registering the first time, with every hit adding one more owed task. Be sure to do this plenty before a job; you don't want to run out of tasks too early.