LinkedIn Contacts is a power user's dream. Not only is it a speedy little app for searching through all your LinkedIn connections, it also allows you to sift through them with ease. LinkedIn Contacts allows you to filter your connections by tags, companies, titles, locations or sources. This is much easier than trying to manually find that name of a connection when all you've got is, for example, that he used to work for Apple and lives in London. You can also browse through your connections alphabetically, those you've newly connected with or by those you've had a recent conversation with (meaning, you've sent them LinkedIn messages).
The app also has a nice "To Do" function, which shows you the job changes and work anniversaries of your connections. As a journalist, this is very handy because I can easily see when one of my PR contacts changes to representing a different company. Finally, the app also offers a calendar function, where you can import all your events from your iOS calendar and combine them with LinkedIn's data. For example, if you have a meeting set up at 2 PM with Sue and Josh from Apple, you could quickly pull up their LinkedIn profiles to get the latest information about them.
LinkedIn Contacts is a free download.