Whether you need a router for your iPad depends on your iPad model and how you intend to connect it to the Internet. Wi-Fi-only iPad models require wireless Internet service -- so if you currently only have wired broadband, you will need a router. Wi-Fi + 3G models can use cellular connections rather than Wi-Fi, so you can connect to the Internet without a router.
A router converts broadband Internet signal into wireless Internet signal. This means you are using your standard Internet connection method, such as DSL or cable, and can use the features your Internet service provider allows just as you would on your wired connection. Wireless connection through your router will be slightly slower than wired connections, but it will still be high speed. A 3G connection uses cellular data towers instead of cable or digital lines, and it is subject to the limits of those lines, including coverage area and data transfer speed.
Even with a 3G iPad, a router is a good idea. Wi-Fi data transfer speeds are faster than 3G, so your Internet access will be faster with a router and a Wi-Fi connection. If you plan to download or stream video, a router is particularly important, as video files require transferring large amounts of data at high speeds. Some 3G plans also limit your data allowance each month, while broadband Internet access through Wi-Fi generally features unlimited usage, depending on your Internet service provider.
A wireless router will cover your iPad Internet usage in the area where it is set up. If you tend to use your iPad when you are away from the home or the office more so than when you would be within the router's range, a router may not be necessary. In this case, you might use a different computer on your home or office network and use the iPad at public hotspots or with a 3G connection while away.
Adding a wireless router to your Internet service allows you to control your network's security settings rather than accessing the Internet through public connections. Wireless information is always at higher risk than wired information, but setting up your router with encryption and password protection can help you keep your data secure.