The Bitcoin software if “Open Source.” This means that anyone can read the software and anyone can participate in the development of the software. The software is found at https://github.com/bitcoin/bitcoin.
The original code was supplied by someone who called himself Satoshi Nakamoto but the actual identity is generally unknown. Some have suggested that since the person is unknown then the software could have “back doors” or some other issue. However, the software gets reviewed and tested by large numbers of people.
The developers have some level of control over the “official” release. There are a number of complex arguments about how much power they actually have since most people will install the default software. However, anyone can create their own release and distribute it to anyone who wants to use it. If a release changed the basic rules of Bitcoin a different release could be prepared by someone else. Users would decide which coins to accept as they would not be compatible so coins created in one system could be spent on the other system. This is known as a “fork” and there would be essentially Bitcoin1 and Bitcoin2. Even though there are many disagreements about the parameters of the software there has been no substantial movement to “fork” the current Bitcoin system due to proposed changes.
Many developers have “forked” the Bitcoin software to create a completely different type of currency called alternate coins or “alt-coins.” There are large numbers of these coins that change different parameters such as the number of coins, the time between confirmation, different algorithms for Bitcoin mining, and so forth. See CoinChoose.com for a list.