Business Intelligence Development Studio, aka BIDS, is actually plain old Visual Studio. When you start Visual Studio for the first time, you can choose a 'role'. For example, .NET developer, project manager or BI developer. Visual Studio then adapts its user interface to make it easier for that role to use it.

In short, download Visual Studio (or Visual Studio Express for free) if you want to develop SSIS packages.

You don't need BIDS to run packages in SQL Server and you can always create simple packages with the data import wizard in Management Studio, but for more complex packages, you need BIDS.