There is only one akathist that is appointed to be done in the services of the Church, and that is the akathist of the Mother of God, which is done on the 5th Saturday of Lent. However, there are many akathists that have been written for private use. Russians in particular love them.
One big advantage that they have is that you can do a substantial service for various commemorations, and you don't have to worry too much about how to go about putting them together.
But how do you do an akathist at home? There are more elaborate ways to do it, and simpler ways. You can combine an Akathist with a canon, or several canons. In fact, as part of preparation for communion clergy (and many pious laity) will do three canons and an akathist. The least complicated way to follow that practice is to get the text all laid out, rather than put them together on the fly. Holy Trinity Monastery in Jordanville, NY, publishes a booklet for this purpose. You can also find that text laid out in their Horologion.
For the complicated order of service for doing an Akathist at home, you can find the outline in the back of the Jordanville Prayer Book, or here. However, a simple way to do an akathist is simply to take the rule of St. Pachomius, and instead of doing the Jesus Prayer 100 times, do the akathist you wish to do at that point.
Not all akathists can be sung using this melody, but most akathists are written following the pattern of the akathist of the Mother of God, and so you can use the same melodies that are done for those too.
The common way that this is done in Russian practice can be heard in this video:
There are many akathists that are available online, but there are two volumes of them published by Jordanville:
Akathists are especially useful if you are unable to attend services for a feast or the commemoration of a saint, either because of distance, or maybe because your parish has no services for a particular commemoration. There are many akathists that have not been translated into English, and many are being translated all of the time... and new ones are also being composed, and so if you look hard enough, you likely can find an akathist for most major commemorations, and if there isn't one out now, there may be one in the future.
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Akathists and Canons Online