The three erasers in Paint by Zaweba and when to use each
How the Eraser, Background Eraser and Magic Eraser in Paint by Zaweba work, with Erase to history, and why masks are often the better choice.

Rubbing out is the oldest editing job there is, and Paint by Zaweba gives it three tools: the plain Eraser, the Background Eraser and the Magic Eraser. They look similar on the toolbar but behave very differently, and choosing well saves a lot of tidying up afterwards.
The Eraser
The Eraser removes pixels where you paint, leaving transparency on a normal layer. It has three modes. Brush mode erases with a soft or hard brush from the brush engine, including its pressure and shape, so it feels just like painting in reverse. Pencil mode erases with hard, aliased edges, the natural partner of the Pencil tool for pixel art. Block mode erases with a square block of fixed size, which is the quickest way to clear rough areas.
Erase to history is the less obvious option. Instead of removing pixels, it paints back the image as it was at an earlier point in the History panel. That makes it an undo brush: you can bring back part of the picture from before a filter or a heavy edit, while keeping the change everywhere else.
The Background Eraser
The Background Eraser removes colour based on what is under the centre of the brush. Sweep it around the edge of a subject on a plain background, keeping the centre over the background, and it removes the background colour while leaving the subject's edge, even inside the brush circle. It is the tool for hair against sky, a product against a coloured wall, or a logo on a scanned page.
Keep the centre of the brush on the colour you want gone. If it drifts onto the subject, the tool samples the subject's colour and erases that instead.
The Magic Eraser
The Magic Eraser is the one-click version. Click a colour and it removes that colour, and colours like it, in one go. A logo on pure white, a scan with a flat background or a graphic with a solid fill are ideal. Busier backgrounds with gradients or shadows tend to leave patches, and the Background Eraser or a selection does better.
A note on 32-bit images
The Background Eraser and the Magic Eraser are refused on 32-bit documents. They depend on comparing colours to decide what to remove, which doesn't work cleanly on high dynamic range data. Convert to 16 or 8-bit first if you need them, or use a selection and mask instead.
Erasing versus masking
Erasing is permanent. Once pixels are gone, the only way back is undo or Erase to history. A layer mask hides pixels instead of deleting them, so you can paint them back with white at any time. For cut-outs you might want to adjust later, such as a product shot that could go on several backgrounds, a mask is nearly always the better choice.
Paint by Zaweba makes masks easy to work with, and its selection tools and one-click cut-outs can usually get the edge right faster than erasing by hand. The erasers are still the quickest answer for small jobs: cleaning stray marks from a sketch, removing a solid background from a logo, or tidying the edge of a layer in a collage.
Getting clean edges
Zoom in and use a soft brush at a size a little larger than the edge you're working on. Check the result against a contrasting background colour layer placed underneath, since fringes that are invisible against white often show against black. If a fringe of the old background remains, go round the edge once more with a small, soft brush at low opacity rather than one heavy pass.
Work on a duplicate layer when you're unsure, and hide the original. If the erasing goes wrong, you can delete the copy and start again without losing anything.