Starting today, Instagram is bringing direct messaging options to the web. As noted by The Verge, a small number of Instagram's global users can access their direct messages from the Instagram website in addition to the iPhone app.
The rollout today to a limited number of customers is just a test, though depending on how it goes a wider rollout could come in the future.
Direct messaging on the web is almost identical to direct messaging through the iPhone, so the interface will be familiar to Instagram users. You can start a DM with someone from a profile page or the DM screen, and there are options to create groups.
Double tapping likes a message, there's an option to see how many unread messages are available, and photos from the desktop can be shared in DMs. Notifications for DMs can also be received if notifications are enabled for the Instagram site in the browser.