There are two primary means of joining a Coalition: The Academy and the Traditional Request.
The Academy is the quickest means of joining a group. That doesn't mean it's the best way or that the group you will get into is any good. For you to join an group via the Academy, that group's Hegemon must have turned on their Academy option and their Academy must have at least one open slot (out of the 10 available). As long as those two conditions are met all you need is to be level 5. You then find the group you want to join via the Coalition Listings in the Embassy and click "join". That's it. Congratulations! You are now a member of XYZ Coalition....sort of. See what happens when you join is that you get the badge for being in the group but, you start at the lowest possible rank and have no other connection to the group...yet. What you should do next is contact the Hegemon or other Officer of the group and introduce yourself. Demonstrate you are a valuable addition and you will no doubt be promoted to full fledge member in no time. If however, you never reach out to the group...you stand a good likelihood of simply being cut.
The Traditional Request
This can actually happen one of two ways. Either you can request to join the group (via the Embassy if you meet the requirements AND they don't have an open and active Academy) or Leadership of the group can send you an invite. Once the other party confirms your application (you if it was sent to you, them if you requested), you will join the group at one rank higher than an Academy Trainee...a recruit (it's not much but it beats being a trainee), you will then have to prove your worth and meet the groups criteria for advancement. Typically the minimum level for joining a group the traditional route is level 16 but it can be much higher than that depending on how high a demand/standards the group has.
You can only be a member of one Coalition at a time but you may apply (if going the traditional route) to multiple groups at a time. A Hegemon that accepts the request of someone who is already in another coalition will see the request removed from their incoming requests but the individual will only temporarily appear to be added in group. Refreshing the page will show that the player was not actually added and the Hegemon can accept another application in it's place.
Consider messaging the Hegemon before applying. State why you feel you would be a good fit for their organization. Coalitions have only so many slots available. Most coalitions worth joining will have some standards on who can join above the simple level restriction. For example, they may require raiding, daily activity, friend requests, minimum troop count, etc. It's a good idea to find out these things first and then apply if you think you still qualify. You can send a message to a Coalition's Hegemon through the Coalition's entry on the coalitions tab of the Embassy.