How to change careers?

Currently I work for a company I don’t care for and more importantly in a job I’m not crazy about. It’s very analytical which is something I’m not, and honestly working so that I can make the company more money but I don’t feel like I’m doing any good for anyone but the company’s bottom line isn’t something I’m interested in anymore.

So I’m interested in changing careers – while still using some of the skills I’ve built up in the business world but go into something non-profit/public orientated. How do I explain my desire to leave the private sector in cover letters/applications without sounding bitter towards the atmosphere I’m hopefully leaving behind (because I know that no matter what that won’t sound good).

So any suggestions for how to switch from one career to another (also any suggestions regarding what careers to go to or stay away from) would be appreciated!