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For some phones, you would just plug into your computer and it would show up as a removable drive. You would then drag and drop the files you wish to transfer between the device and computer. If some of the files don't play natively on your Android device you would have to search for an app that will play that media format. If it is an iPhone, you would use iTunes to sync the media.
If your phone has a removable micro SD/memory card (not all phones do) you can transfer content from your computer directly onto this card and reinsert it into your phone. You will need a mirco SD card reader in order to use this method.
On android: If it doesnt pop up on your computer in a moment, check for a notification giving you options on the phone itself. You might have to set it to connect as "USB Storage" or something similar. For an alternate solution, a real handy app to transfer information on android phones is AirDroid. This app lets you connect your phone via wifi and view everything on your phone in your web browser.
For older phones, transfering media such as your photos is often only possible when the media is stored on a micro SD card. Alternately, if you have a smartphone as well you can try pairing to that phone via bluetooth and sending the pictures to the other phone that way. They might be more accessible once on a newer device.
You can use an application such as dropbox. Install dropbox on both your phone and computer (dropbox.com) then all you have to do is drag files from your computer into the dropbox folder. Once your devices are synced (you would log in with email address and password) whatever is in the dropbox folder will show up on all of your devices.