Personal Hotspot is your LTE connection being relayed to another device. Browsing the web for example, when connected to a mobile hotspot will result in the usual data usage or data charges.
Wifi is your standard wireless internet connection, which deals with your ISP.
Cellular is the ability of a device to use cellular data connections, like 3G, 4G LTE, etc.
iPads (2nd gen and up I believe) can use airplay. You must be connected to the same internet network as the Apple TV, and then you pull up the control center, select airplay, then select the Apple TV you want to stream to, then turn on mirroring. Depending on your settings, you will either connect automatically, need to enter an on screen code, or enter a pre set password. (On iOS 6, you need to double click the home button, switch to left most page in the multitasking panel, select the airplay symbol, and carry out similar steps afterward.)
Files will usually automatically sync with iCloud.
So anyway, a wifi+cellular iPad has the ability to connect to wifi and cellular (3G, 4G LTE) networks, similar to an iPhone. Whereas a wifi only iPad can only connect to wifi networks.