You need to build enough of a preview that you, and your initial beta users, are able to tease this app out to others driving them to want an invite. That’s the beta method and really it works best when run through viral means; the more people talking and sharing, the more people you get and the more people trying to get in.
Of course if you don’t have the network you will need to build it, and quickly. Put together a 30 second pitch, simplify it to a basic site, and start using the appropriate networks on twitter, linkedin, startup communities and related sites to get people looking and sharing. If what you have is compelling people will respond.
You can contact bloggers, ask them if they want an invitation and if they are willing to review your service in their blogs, and give them invites for their audience too. I do this for a while and it had great results.
Also you can allow your users to invite their friends
I have to agree with Ted. I’ve never understood why some Apps/sites have those closed-door invites where you’re left feeling confused as to whether or not you even want/NEED to sign up. Is there a Video that shows the BENEFITS of your app? That’s the key word right there: Why should users give you their info and sign up for, yet another, app?
Show them what you provide and market it to others via video, screen shots, etc. People love visuals. Give them enough to WANT to sign up.
Who does the app benefit? Find that crowd and start posting in forums like this (but focused on that group), find Google Groups, use PPC, etc.
IMO, using paid traffic will be, by far, the quickest way to get people TO the site. But how do you turn those clicks (PPC, etc) into CONVERSIONS? Once they arrive at your site you need to convert them into sign-up’s, right? So make sure that your Landing Page has the info people want (text and visuals).