So the first thing you need to know to create different animations in Blender, is what are the actions? How to create one? How they work? and How do you delete them?
An Action in Blnder is a tool to record and contain the data. As everything else in Blender, Actions are data-blocks. So when you animate an object by changing its location with keyframes, the animation is saved to the Action. That way you can create as many animations as actions you want.
Each Animation could be a different action on the character or object. For example run, iddle, shoot....
Then to view the action menu, change the content of the page to the Dope Sheet and then you can change it to the Action Editor.
Changing the window to dope sheet
Menu to change to the Action Editor
When you have the editor in view, you can start creating actions, but before you do that, you must have a couple of things in mind:
Once you have created and armature (Top corner Add→Armature), you are ready to create actions. With the Armature selected, press the New button, change the name of the animation and don't forget to press the shield (If you do not press the shield, Blender will not save the new name, and you will lose the changes you do with the action).
With the armature Selected create a new action
When you finish with the action, close it by pressing the X button at the right, so you access to the first menu on the editor and create a new one from scratch. An Important note if you want to delete the action because you don't like it, don't need it or if you duplicate it by mistake, DON'T press the X button alone (as stated before this closes the action but it doesn't delete it) and DON'T press the delete key nor the backspace one this won´t delete it either.
The ONLY way to delete it is to press Shift + the X button on the right, then close and reopen the file to refresh the motor, if you do not close the program you will still see the actions that you deleted.
From Maya To Blender
Now let's say, that you want to work in another software like Maya, This time the change is easy, but you have to follow a series of steps.
Once you import to blender; the fbx will create an armature in the scene with the animations in it. Then all you have to do is :
Exporting and Viewing
When you export the model, check a few options in the presets:
Web with the different animations to select