Creating new layer from selection in QGIS?

After making a selection in a layer, right click on it and choose save selection as.

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Update for QGIS 3.2 Bonn as of 2018

As this is the accepted answer and the way of doing this has evolved somewhat, beginners might find this answer incomplete, so:

After making a selection in a layer, right click on it and choose "Export" > "Save Selected Feature As..."

which is actually just a nice redundancy, because it takes you to the same mask as clicking "Save Feature As...", but it already checks the box "Save only selected features"

A new feature just came up in QGIS 2.2 called "Paste as new vector layer":

It is a common activity in a GIS to create a sub-selection and then to create a new layer from the selection. In QGIS you can already do save selection as to save a layer from your selection, and now offers functionality that allows you to create a new file or memory layer from whatever is in your clipboard. Simply select some features, copy them to your clipboard and then do Edit -> Paste features as and choose either New vector layer or New memory layer from the submenu. The best part of this new feature is that if you have some Well Known Text (WKT) features in your clipboard from another app, you can simply paste them into QGIS as a new layer now.

In QGIS 2.6.2 Brighton, you can do the following:

1. Select features
2. Right click on the layer
3. Save as...
4. Select the option "Save only selected objects"
5. Save

That should do