An ally is another TE player with whom you are "allied". To become allies, the two of you stand next to each other and wait for the offers to come up. An offer says "do you wish to ally with x?", you say yes, they get the offer, and say yes or no. It's the same basic process as changing allegiance. You both have to be Count or higher. (One word of warning: if both players are at the Prince level, the huds may get stuck on offering to let one join the other. In that case, detach and re-attach.)
So now that you've made an ally (or know that you can), why should you? What good is one?
- If you are sabotaged, an online ally will get the chance to pay to help get your acres back. (This is often seen in the announcements as 'an investigation funded by one ally convicted the main suspect...')
- You can transfer citizens between each other. This transfer becomes even more efficient if you build an embassy
- If you build four embassies, you get an extra gold per acre on your land.
One note about embassies: they are expensive (roughly five times acre cost) and they take an acre of your ally's land (your ally loses an acre of personal land when you build one.) On the other hand, you start getting good offers on your land which is attributed to the embassies. And the cost of an embassy can be reduced if you buy it during a year of cheap land from the trader. That year adjusts your land price down to a third of its normal rates, and a lot of calculations- including loyalty rewards and embassy costs- go down as well.
Since you can have up to four allies, and since you get that extra gold per acre after building four embassies, people who get the federation hud are strongly encouraged to get allies as quickly as possible- preferably ones who have plenty of citizens to give them!
But when looking for an ally, you must keep in mind the fact that you can not have your liege or a direct subject as your ally. This means that when you want to make allies within your own group (a natural desire), you have to look a bit further afield. It may be your liege's liege (or a subject's subject). Or it may be another subject of your liege. When trying to track who is allied with who, and who is lieged to, the joins can quickly become an interlocked mesh of two separate webs of connections.
Since things can be so complicated, and since new Federation HUD players are encouraged to get allies soon, they often find friends outside of their immediate group- may, in fact, have to. And why not? I know that when I find that there are people in my group who have allies elsewhere, I tend to say "great, I'm glad to hear it." It's a nice way for friends who are lieged to different areas- even kingdoms- to still share a friendship within the game. It's a good way to find other people (or have your liege find other people for you!)
And, in the end, that may be the best reason to have allies: to recognize existing friendships and create new ones.
Who needs allies? Perhaps the question might be better asked, who needs friends?