Trello.com is more of a really cool lists of lists application. They have iPhone app as well as the website works great on mobile web browsers. We have experimented with it on the project at work to lists up and make improvements on how we run the project. A key point of the agile scrum methodology is to constantly improve the process. By quickly adding suggestions on improvements into Trello we have managed to cut down on the time spent on standup meeting and identify different improvements we can do regarding the source code.
YouTrack is a free application, up to 10 users, from JetBrains. I tried out version 3, and it worked more as a bug tracking tool for me. A bit different way of working where you can view different tasks from a powerful search bar alone. It did not include the famous states tasks can be in. States like: To do, in Progress, Testing, Fixed and Done. At least I could not find them.
But now JetBrains have come out with version 4. And the lanes, for the different states a task can be in, is one of the first thing you see in the "whats new" section of the website.
If you need to organise something into a lists of list: use Trello. If you need a serious bug tracker or a lightweight Scrum/Kanban tool to manage you projects: use YouTrack. If you want to spend money and use the best of the best: use Jira