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I have got a number of polygons within a single larger polygon. Parts of the smaller polygons go out from under the larger one, and I want to delete the parts of the smaller polygons, that go outside the larger one. How can this be done?

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    Welcome to Geographic Information Systems! Welcome to GIS SE! We're a little different from other sites; this isn't a discussion forum but a Q&A site. Your questions should as much as possible describe not just what you want to do, but precisely what you have tried and where you are stuck trying that. Please check out our short tour for more about how the site works.
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You want to clip (QGIS -> Vector -> Geoprocessing Tool -> Clip) your smaller polygons with your larger polygon. This tutorial describes how to do this.

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If your dataset is simple enough - one large polygon, you could use the clip function (menu Vector>Geoprocessing>Clip). First select the large polygon and export it into a new layer and use that as the overlay layer.
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It also sounds like you might be able to use the Delete Part tool. For example, I have a countries.geojson. In it the United States polygon includes tons of tiny islands near Alaska. I want these gone and only the US mainland to remain. The Delete Part tool allows you to remove any of the marked polys by clicking on them (I already deleted Alaska itself). The tool will work on polygons both selected and unselected.

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