Learn Spanish (Latin America): No tires el peluche al piso. - Do not throw the stuffed animal on the floor.

Because this is a negative command telling someone not to do something. For regular -ar verbs like tirar (to throw), the negative tú command changes the normal ending from -as to -es. So, tú tiras means 'you throw', but no tires means 'don't throw!'.

Can I just say No tira to tell someone not to throw it?

No. No tira is just a statement meaning 'he or she doesn't throw' (or the formal 'you don't throw'). To give a negative command to a friend or child (someone you call tú), you must use that special negative command ending, which is -es for -ar verbs.

Why does it use al (to the) instead of en el (on the) for 'on the floor'?

In English, we say we throw things 'on' the floor, but in Spanish, the action of throwing implies motion toward a destination. Because of this movement, you use a (to). Since piso is masculine, a combines with el to form the contraction al piso (literally 'to the floor').

Does tirar only mean 'to throw'?

It means 'to throw' or 'to toss', but it is also commonly used to mean 'to throw away' (in the trash) or 'to pull' (like pulling a door open). Here, the context of the floor tells us it just means dropping or tossing the stuffed animal.