The Embassy allows the formation of alliances. It is also a prerequisite to building the Space Station. Each level of the Alliance Leader's highest-level Embassy allows 2 more members to join the alliance. Alliance members need only have a level 1 embassy to participate.
Stash: The stash can hold a total of 500k resources for the entire alliance. Any member with an embassy may donate as much and as often as they like up to this total limit on the stash size. In addition, each level of a member's embassy allows 1 stash exchange per day, and 10,000 more resources per exchange. This exchange limit is planet-specific for each embassy, so having a high level embassy on every colony allows for free and instant exchanging of resources. For instance having a level 10 embassy on every planet allows for the possible exchange of up to 1 million resources between any two planets in your empire at no cost and no travel time. (Example: Planet 1 does a 100k exchange of resources that Planet 2 would like for resources already in the stash. Planet 2 then does an exchange for the 100k of resources added by Planet 1 putting back in resources helpful to Planet 1. Repeat 9 more times.)
Warning: Dissolution of an alliance will abandon (and destroy) all space stations (since the alliance is a prerequisite of the SS).
Once you're a member of an alliance you get a couple cool things. First, you can email all your allies by typing @ally into the "To" field in the in-game email system. Second, you can passively use all the probe data of your allies. This means you will see the bodies in a star system an ally has probed if you happen upon it on the star map, but there is no direct way to get a list of your allies' probed stars.
You may find yourself in a situation where you built an embassy on your original home world and formed an alliance there. Later on, you may want to relocate your embassy to a colony to utilize the embassy's plot on your home world.
Relocation of the embassy for alliance members isn't bad at all. As long as you have an embassy, you're not prevented from demolishing any extras.
It should be noted that you will be unable to demolish any embassy on any of your planets so long as you're in an alliance. So, the correct relocation process should involve: dissolution of the current alliance, demolition of the embassy to remove, reformation of the alliance at the new embassy. Reforming the alliance on the new planet before demolishing the embassy on the old one will prevent the demolition of the old embassy.