Well, continuously charging a lithium ion battery can shorten it's life. So it is generally not recommended that you keep it plugged in all the time. The same can be said for laptop which is why some laptops only charge the battery between 50% and 80%. You must override the setting to get a 100% charge.
The battery does not get continuously charged. Once the battery reaches 100% the circuity should automatically turn off the charging state. However, the circuitry "tops off" the every now and then to maintain the 100%; known as trickle charge.
Everyone wants a battery to give the most battery life they can get. They want their electronic device to last a long as possible on a charge. Ironically, this is actually what shortens battery life. The longer the battery is forced to hold a full charge the quicker the battery degrades (hence why many laptops automatically limits the battery charge between 50% - 80% capacity). Without getting too technical, the high voltage eventually causes the lithium to "crystalize" forming "lithium plating". Once the lithium in the battery changes to this state the battery begins to loose it's capacity; meaning the battery does not last as long as when it was new.
With regards to batteries bought online, some could be manufactured cheaply especially if the battery is being sold for an inexpensive price. These cheaply made batteries tends to loose their charge more quickly compared to more expensive replacement batteries. I would suggest buying replacement batteries from Amazon and read the user reviews. If too many people complain about the quality of the batteries, then avoid them.