Raster: the image is made up of tiny coloured squares which map to individual pixels on the screen when the image is displayed at a scale of 1:1 but if you scale it up to look bigger then it gets blurry. You get raster files with paint packages like Paint.net and Photoshop.
Drawing packages that use brush tools to draw with, but save in a raster format (gif, png, jpeg) immediately lose all the information about the type of brush, so typically to edit it again you need to change each little square of colour, or draw over it - or start again
Typically raster is the type of format you save as when you've finished your drawing and want to use it in software or on the web.
Vector: the image is made up of brush strokes and fills, and that information is retained so that no matter how much you scale the image up and down, the image is drawn with the appropriate size brush and you don't suffer from loss of quality. You get vector files with Illustrator and Inkscape.
Drawing packages that save in a vector format remember all the information about the type of brush so that you can change the colour or thickness of a brush stroke next time you load/edit it.
Adjustment layer: applies a special effect to your image. Imagine holding an orange trnsparent sweet wrapper over your image and looking through it - you get an orange tint to your picture - the sweet wrapper is an adjustment layer. If you scrunch up your sweet wrapper you might get a crinkly effect as well. You can get all sorts of different special effects.
Mask: imagine cutting out a shape from the middle of a piece of paper and laying the sheet with the hole over the top of your picture and painting over it so that you end up really only painting over the bit that you can see through the hole. Well the piece of paper is basically a mask.
You can combine your stencil cutout (mask) and sweet wrapper (adjustment layer - or just 'layer') together if you wish. For example if you combine sweet wrapper and stencil you can apply the orange tint effect only to the area through the mask.
Adjustment layers and masks are not saved as part of a raster file, but are part of the drawing packages custom format file, so that you can change it when you load it back in.