If you are talking about changes to the user interface, then anyone can grab a copy of any of the existing open source user interfaces and make any changes they like. They can then share their new user interface with whomever they like.
If you are talking about changes to the blocks or transactions, and acceptance and rejection of those which leads to network consensus, then you are talking about a "forking" change. While you (or anyone else) are welcome to modify the existing bitcoin programming, any changes that violate the current consensus rules will result in all peers that are running the current protocol to simply ignore such communications from program.
You could potentially convince other peers to run your protocol changes. If you were successful in getting a significantly sized minority to run peers that make use of your changes, then the system would split into two separate systems. Those running the "legacy" bitcoin, and those running the "new" bitcoin. All peers on the "legacy" system would simply ignore the communications from your "new" system.
If you can convince a significant majority to run peers that make use of your changes, then the system would still split into two separate systems, but since the majority are using the new system the legacy system will be more likely to be seen as the niche eccentric system, while your new system will be the "mainstream" system.
Currently, the easiest way to get a significant majority of peers to use your software changes is to convince Wladimir J. van der Laan to include it in the "Bitcoin Core" reference wallet. You could make the changes yourself, or get someone else to make them. You would participate in discussions with some of the other programmers that have provided significant amounts of programming effort to "Bitcoin Core". If everyone agrees, then Wladimir J. van der Laan will pull your changes into his github repository. If there isn't agreement, then Wladimir J. van der Laan might choose to wait for more discussion before making the decision to pull the code into his github repository.
Potentially, Wladimir J. van der Laan could refuse to pull in the changes at all, or could put off the decision indefinitely. If either of those things happen, you would have to find some other way to convince everyone to use your software instead of the software that Wladimir J. van der Laan has control over.
Thanks for all the answers, but this one had the exact info I was looking for. P2P consensus, albeit having been around for a while, is still a relatively new concept when we're talking about such a large scale as a currency, so something so obvious still needs a kind of "rewiring" of traditional thoughts in order to be understood. I tried explaining this to some people the other day, but the recurring question was - who is responsible for BTC? I answered "no one and everyone", but so far it is still a difficult concept to swallow.
Myself and another enterpreneur are currently in charge of spreading the word about Bitcoin in Malta, as he took the initiative to start monthly meetings and I liked the idea of gathering around BTC enthusiasts. Still, ignorance about BTC is still extremely high here, so we're aiming to double our efforts during the summer