A ginger ale plant is simply the trub left over from making ginger ale. You use it to start your next batch or divide it up and give it to friends to use along with the recipe. It's called a plant because it's a living thing. It shouldn't be called beer because it's not made with hops.
I make a lot of traditional English ginger ale in the summer using ginger, lemon juice and sugar. Some people add other ingredients but that spoils it in my opinion. I make a starter plant using bakers yeast which adds a delicious fresh bread taste. I've never used beer yeast so I wouldn't know which variety to use.
I often make it to about 6% ABV but sometimes less. It's a wonderful drink which is very good for the stomach and aids digestion. It's great for BBQs and most people love it, especially the ladies. It's not hard to make and all the ingredients are available from the supermarket.
The only problem is that it's so delicious and easy to drink that it disapears too quickly and people get drunk very easily because they don't realise it's alcoholic. They think it's a soft drink and tip it back like lemonade. They need to be warned.
When I have a brew day I sometimes make batch of ginger beer for 'her indoors' whilst I'm waiting for my beer to mash. I make the starter the day before. That way I keep her happy and we all know how vitally important that is.
Her friends sometimes invite her to BBQs and say "bring some ginger ale". I then get pestered into making it and I can come on all reluctant and say "Oh... alright, I suppose I could make some time for it. I could always mash some beer while I'm waiting for it to start fermenting". It doesn't fool her for a second but she's happy and I'm happy so everyones a winner.
dogstar- anyone is entitled to their own definition of what the ginger beer plant (GBP) is but the OP is referring to the keffir-like grains that are essentially polysaccharide conglomerates made by the weird range of bacteria and yeast that they host. they slowly ferment the ginger beer precursor that you put them into, adding flavor and allowing the fermentation to continue on a bit to give carbonation after they are removed. they are great because the flavors are unique, they don't ferment very quickly, leaving you with a sweet drink, and there is not much cloudy yeast to clear out of suspension. you just strain them out and put them to rest, or add to your next batch. they are also really cool to watch, as the booger-like beads float up and down in the drink while they are active. GBP makes a very different drink from a yeast-fermented ginger beer, usually so low alcohol to be essentially non-alcoholic (~0.5-1% abv), worth a try if that's what you want and you can find the plant. i got mine from the uk link listed above- gingerbeerplant.net so i can vouch for them. in various discussions in the 'soda' forum of HBT there have been (...uninformed..) people saying that that's not GBP but keffir grains and that they know the difference, blah blah. i suggest ignoring that type of argument.
There is a major difference between using the ginger plant scoby and a typical yeast ferment. They are nothing alike. I do both and like both very much. It would be somewhat like comparing sauerkraut or kimchi to fried cabbage, or cottage cheese to yogurt. They aren't the same and aren't intended to be.