Formal politics is politics as defined by law. Essentially, everyone follows the rules. One man, one vote, everyone follows the rule of law, people can petition their government for wrongs and have them heard, and so on.
Informal politics isn’t really defined, but it is when there are other forms of communication channels used that aren’t supposed to be there. In the United Nations, this takes the form of backdoor channels and is considered a good thing. (For example, two blood enemies talk together secretly to reduce tensions or lay the groundwork for an agreement when it would be political suicide for their citizens to know negotiations are going on. Example: Egypt and Israel peace agreement. Lots of informal negotiations preceded the actual peace talks and agreements.)
In Western countries, informal politics serves both to help and hurt the legitimacy of the political process. For example, Senator Daschle is not registered as a lobbiest, but obviously he lobbies. This is legal, but below the radar. This can both help more effective legislation get passed (which is good), but the actual influence is not documented for public review (which can be bad.)
In developing countries, informal politics tends to undermine the rule of law by creating a tendency towards corruption and cronyism. This is especially true where a dictatorship is in place, or a democracy is new. Paying a bribe to get a hearing is an example of informal politics. So is talking to your tribal elder (through family channels) when he also happens to be a judge or senior government official.
