My phone shut down and only the logo

Software has gone corrupt on the phone most likely. Leave it on charge for longer if you haven't left it for more than 5-10 minutes.

Try charging up your phone before performing these steps below.

Check the charger port for any blockage of dust / debris / etc. If there is, try cleaning it with a toothpick or something equivalent.

Disclaimer: You will lose all data in your internal storage on the phone if you follow the steps below.

Steps to Factory Reset most (if not all) Samsung phones:

1. Turn phone off by taking battery out for 30 seconds or force turning it off (if you don't have a user removable battery) via holding power button and volume down button.
2. After that Boot your device into Recovery Mode. You can do this by holding the Volume Up + Home + Power keys. When the Samsung Logo pops up, stop holding the power button but keep holding the rest.
3. Once your android device is in Recovery Mode, Go to the “wipe cache partition” option and select it.
4. After that go back to the recovery main menu and reboot the device by “reboot system now” option.
5. If the device again stuck on Android Boot Loop pull out the battery again or power off by other method specified above and follow the steps above again. This time also “wipe data/factory reset” and then reboot device.

Now your device should come out of the Android Boot Loop and at the first time the device will take some time to reboot.

If these steps still don't work then you may need to restore the software on your phone via Samsung Kies. If I can remember you do it via Tools > Firmware upgrade and initialization. This will prompt you to enter the model number and the serial number of the phone which is usually at the back of the phone or in the battery compartment.

If the Samsung Kies method does not allow you to download the firmware for your phone then you'll have to go to http://www.sammobile.com/firmwares/ where you type your model number of your phone to find the firmware for your phone. Make sure you choose latest version to the network it was mainly with.

Samfirmware instructions (After the file has finished downloading):

1. Extract (unzip) the firmware file
2. Download Odin v3.10.7
3. Extract Odin ZIP file
4. Open Odin v3.10.7
5. Reboot Phone in Download Mode (press and hold Home + Power + Volume Down buttons)
6. Connect phone and wait until you get a blue sign in Odin
7. Add the firmware file to AP / PDA
8. Make sure re-partition is NOT ticked
9. Click the start button, sit back and wait few minutes
10. Before moving to step 11, you may want for it to get to the welcome screen as it sets up the phone.
11. Wait until it says pass. It is then safe to unplug after that.