Firstly, social sites like Facebook and applications like Evernote are very different. If you could be more specific about your "great product", it would help.
As with any product, whether or not people use it depends on whether or not it is useful to them. A social network with no users - so no network - is useless, so getting your first thousand or so is a massive hurdle. You just have to persuade as many people as you can (friends, colleagues etc.) to sign up, and hope that, like Facebook or LinkedIn, it will "ignite" and grow on it's own.
If your website is based on user-generated content (any forum, stack exchange, flickr etc...), then you need to generate as much content as possible on your own. These type of sites are a little easier to start, because they are often useful even if you don't know anyone using it. Having said that, they do still need to have some user base, so again get everyone you know to sign up.
Web apps like Evernote or Google Docs are the easiest to spread, since they can be useful in themselves without any user base. If your product is truly great, people will use it on it's own, then tell others about it. Bear in mind, though, that your product really does need to be exceptional - it will need to be innovative and really stand out to convince people to switch. You don't need a huge budget, just make sure your website looks good, promotes itself well and is easy to sign up for. Then, try sending out your website to bloggers and news sites; it's free, and if your product is decent, they'll write about it and start spreading the word for you.
As far as competitors and copycats are concerned, no one is going to spend the time and money to copy you unless you yourself are getting users and are successful; but if you're successful, you have a head start, and any copycat is going to find it difficult to catch up with you (since they have none of the uniqueness or innovation selling points) without a massive marketing budget.
Finally, a lot of people have tried to do what you're doing now; if you spend some time on the internet, you'll find a huge amount of information about internet start-ups and a lot of interesting marketing ideas.