If I make an app (free) that has ads inside of it, does that count as income and do I have to pay taxes on it?
It's income. It's almost certainly subject to income tax. As miscellaneous income, if nothing else. (That's what hobby income usually falls under.)
If you kept careful records of the cost of developing the app, you might be able to offset those against the income... again, as with hobby income.
In general, all income is taxable, regardless of the source.
If you living in the U.S. -- I don't think you said anywhere where you live -- then if you are donating this money to charity, you would have to declare the income, and then declare a deduction for the charitable contribution. At that point the two would cancel out and the net result is that you wouldn't have to pay any tax on the income, but you can't just leave it off your tax return. Well, even if you donated all of it to charity to that you don't have to pay income taxes on it, you would still have to pay social security taxes, and it would still affect your social security benefits when you retire.
If you're saying that the organization receiving the money is itself a charity, as opposed to donating the money to some other organization that is a charity, than you usually have to be registered as a charity with the IRS to avoid income taxes. There are still forms to file to report the income, but you wouldn't have to pay taxes. There are some exceptions to the requirement to register, basically if your organization is very small and for certain religious organizations.