I want to use RemotePC, but it needs an internet connection. I don't understand why; I only want to remotely control my Mac from an iPad, so a local network should be enough… right?
Macs come with VNC automatically, no additional install. If you can trust everyone on your local network and your VNC ports are closed on your router/firewall, then I would just turn it on. There is a check box under advanced remote management or remote desktop options that says something about allowing VNC users to log in. There's an optional password field. I use iTeleport then to VNC into my mac. There are other cheaper VNC programs too, but iTeleport is pretty excellent.
I use www.logmein.com. I am not sure how well it works nativly, but with the $30 ignition app, it works great. The one thing that might not be best for you is that it is web based - you have to do it over the internet - a local network with no internet access would not work.
RemotePC is designed to connect over the internet - i.e. when you are not on the same network. If you want to connect over e.g. a ad hoc network, look for a VNC client such as Screens or iTeleport (my favorites). They are both great.
Both apps connect to a Screen Sharing enabled Mac. iTeleport can also connect via the internet, i.e. even if you are not on the same WiFi, using an extra application which you run on your Mac.
As for remote control - I would recommend either TouchPad or HippoRemote. They also work using VNC, but instead of showing you the screen, they give you a virtual trackpad/keyboard on you iDevice.
ActyMac DutyWatch Remote is a good app for remote control, and provides access from an iPad.