If you go into the Diplomatic overview for any nation (where you can choose diplomatic actions such as declare war, influence, offer alliance, etc.) any coalitions the nation is in will show up on the left hand side of the screen where you can also see if they are involved in any wars, truces, improving relations, etc. You can also see there if that nation is the target of any coalition, and if so which nations are involved in that coalition against that nation. When you're looking at another country in the Diplomatic overview one of the drop down menus is 'Coalition', where there is a button to 'Join Coalition' or 'Form coalition' depending if a coalition already exists against the nation.
From my understanding coalitions function basically like a patchwork alliance. You may not be friends or even allies of the other members you are in the coalition with, but the threat of a mutual target unites you (if only temporarily) for the greater good. When any nation in the coalition is attacked by the country that is the target of a coalition, all other nations involved in the coalition are also drawn into the war. If you, or another member of the coalition, declares war on the coalition's target then all other coalition nations also declare war. The AI will join or start a coalition against you if it is weaker than you and perceives you as a threat, most likely because you have a large aggressive expansion modifier.
I don't know if you can encourage other nations to join coalitions, but if you can't directly invite someone to join your coalition then you can most likely manipulate other nations through improving relations or making other nations seem more threatening. But honestly, I'm sure there's someone else who would be able to give you more information on whether or not this is a possibility.
Good luck!
