Battery powered MP3 players typically get their power in one of three different ways:
(1) regular alkaline batteries, e.g. AAA or AA.
(2) rechargeable alkaline batteries usually also AA or AA.
(3) lithium-ion batteries such as those found in almost all cell phones.
MP3 players with no battery compartment are most certainly powered by a lithium-ion battery. An example of this is Apple’s famous iPod line of MP3 players. These batteries are designed to be recharged and used throughout the life of the device. This is great for the consumer because lithium-ion batteries are typically going to give you a longer battery life than alkaline batteries.
As mentioned above, it is usually not necessary to change batteries, in the device unless the battery has reached the end of its life which is not the case with your MP3 player. Lithium-ion batteries in MP3 players are typically charged in three different ways:
1) Through a charger that plugs into a wall outlet and connects to the player. Buy a charging kit made specifically for your MP3 player. Some manufacturers like Sansa make charging kits for specific MP3 players. The kits are available at major electronics stores. They come with a charger that plugs directly into the outlet. You plug the USB cord into the port in the back of a charger and then into your MP3 player.
2) Through a USB (or FireWire) cable that is used to connect it to your computer. Get a USB wall charger. These are AC adapters that plug directly into the outlet. You then plug the USB from your MP3 into the port on the adapter. Since it sounds like you did not receive a charger with your MP3 player, your MP3 player probably charges via a USB cable that came with the device. Plug the USB cable into your MP3 player and the other end into your computer and you should see a message that the MP3 player is charging
3) Charge your MP3 player on the go. There are also charging kits that plug into the cigarette lighter or power plug in your car. These kits come with a DC power adapter that has a port on the end for you to plug in your MP3.

