Like a locked chest that only you can access? You can't. You can, however, make an Ender Chest, which is basically a global chest that you can access from any Ender Chest. Only you can access your own Ender Chest inventory.
The same way you do on singeplayer - right click the chest. However, if you are referring to a bukkit server or any server with plugins and the chest is locked, there is no way to do it yourself. This is to prevent theft and it works very well. Additionally, there may be server lag. This means that there may be a delay before the chest items appear. Do not worry, the lag will go away soon (hopefully)!
saving
Open the chest and then you will be able to drag items into the chest, in which they will then be removed from your inventory.
you cant
The Cavalier chest is a type of hope chest that was first made in the 1940s. The best way to unlock a Cavalier chest without the key or combination is by contacting a locksmith.
Click the screen while your looking at the chest - it should open the chest. To get out just tap away (out of the chest)
Yes, actually you can! As long as you're playing multiplayer. :P just type /lwc into the chat box thing to get a list of commands, or if you just want to lock your chest without a password and only you can get into it, then just type /cprivate and left click the chest. :) Also works on other things! (doors, trapdoors, crafting tables, furnaces, etc...)
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You right click on it.
It holds items like a chest
Sometimes it is lag. But in this case, it seems like you have a block on top of your chest. If you put a block on your chest, the chest wont open.