Pushover is a simple application that allows you to receive Push notifications for a variety of services. Thew app of its own can not really do anything, but when coupled with other online services it can become a really powerful tool.
Too get started, you need to install Pushover. Following that, your first stop should be "If this, then that" (IFTTT). IFTTT allows users to create 'recipies' to help automate their online life - particularly their interactions with social networks. For example, if you create a new blog post, you can automate your services to automatically it posted to Facebook, a Facebook Page, Twitter, Linkedin, Tumblr, and then have the entire post backed up on Evernote. this is all possible with a few simple clicks!
Where Pushover comes in handy is for receiving push notifications. You can set it up to notify you of almost anything you want - the days weather, when it is going to rain, when you get a DM or @Mention on Twitter, of your Google calendar events, etc.
You can create your own 'recipies' to utilise Pushover, or browse a large database of user-generated ones.
To give you a better idea of what Pushover is capable of, here are some some screenshots from IFTTT, showing a number of user created 'recipes':
In the words of the developer:
Pushover - Instant Push Notifications