The Coordinate Reference System Selector allows you to set the projection for the current layer.
You can find a coordinate system by entering its name, EPSG code or the QGIS ID into the text field. The list of recently used coordinate systems are available at the top for quick access.
When operating across layers, for example, computing intersections between two layers, it is important that both layers have the same CRS. To change the projection of an existing layer, it is insufficient to simply change the CRS in that layer's properties. Instead you must save the layer as a new layer, and choose the desired CRS for the new layer.