Q: MacBook Air 13 inch fan loud???

This is a user computer, yes? Did you wipe the drive in order to install your own copy of OS X and your own software or are you using OS X that came on the computer from the seller? I'm guessing the latter, so see: Runaway applications can shorten battery runtime, affect performance, and increase heat and fan act… and,

Pre-Mavericks

Open Activity Monitor in the Utilities folder.  Select All Processes from the Processes dropdown menu.  Click twice on the CPU% column header to display in descending order.  If you find a process using a large amount of CPU time (>=70,) then select the process and click on the Quit icon in the toolbar.  Click on the Force Quit button to kill the process.  See if that helps.  Be sure to note the name of the runaway process so you can track down the cause of the problem.

Mavericks and later

Open Activity Monitor in the Utilities folder.  Select All Processes from the View menu.  Click on the CPU tab in the toolbar. Click twice on the %CPU column header to display in descending order.  If you find a process using a large amount of %CPU, then select the process and click on the Quit icon in the toolbar.  Click on the Force Quit button to kill the process.  See if that helps.  Be sure to note the name of the runaway process so you can track down the cause of the problem.

I suggest you start by doing the following: What to do before selling or giving away your Mac - Apple Support. This is what the seller should have done. This will get you properly started with it. Note that the seller needs to remove it from Find My Mac if he/she enabled it. The seller needs to remove it from his/her list of registered products so you can register it on your list: Your Support Profile for Registered Purchases. If you are unable to register it because it is already registered, then the seller needs to remove the registration from his/her list.

Once everything that should be cleared from the seller's stuff, then you can wipe the drive and start new:

Install OS X Using Network Recovery

Be sure you have backed up your files because the following procedure will remove everything from the hard drive.

Boot to the Network Recovery Server:

Restart the computer and after the chime press and hold down the COMMAND-OPTION- R keys until a globe appears on the screen. Wait patiently - 15-20 minutes - until the Recovery main menu appears.

Partition and Format the hard drive:

1. Select Disk Utility from the main menu and click on the Continue button.
2. After DU loads select your newly installed hard drive (this is the out-dented entry with the mfgr.'s ID and size) from the left side list. Click on the Partition tab in the DU main window.
3. Under the Volume Scheme heading set the number of partitions from the drop down menu to one. Click on the Options button, set the partition scheme to GUID then click on the OK button. Set the format type to Mac OS Extended (Journaled.) Click on the Apply button and wait until the process has completed. Quit DU and return to the main menu.

Reinstall OS X: Select Reinstall OS X and click on the Continue button. Be sure to select the correct drive to use if you have more than one.

Note: You will need an active Internet connection. I suggest using Ethernet if possible because it is three times faster than wireless.

This should restore the version of OS X originally pre-installed on the computer.