I have files, in Tiff format, downloaded from ASTER-GDEM.
Can I open this file with QGIS? If yes, how?
From your question, I'm assuming you are a new user of QGIS.
That all being said. Here is how to add a Raster (Tiff, in your case) in 2 easy steps.
Here is the result:
Since your raster is a DEM, there are a couple additional steps necessary to make it display properly.
Here is the end result with the DEM symbolized based on the raster values.
This process all being said and done, I would strongly recommend you to peruse the latest QGIS documentation: Version 1.8
It is likely that these two resources may lead you to an answer, as well as provide you with additional information.