In cookbooks they sometimes mention "plain" chocolate? What is that?
Plain chocolate is dark chocolate, ie not milk or white chocolate. There can be quite a variance in the quality of plain chocolate sold so it is worth checking on the packaging for the cocoa solids content in the chocolate. For baking a good quality chocolate with a high cocoa solids content is preferred for a good flavour - Nigella uses chocolate with 70% minimum cocoa solids.
For the US bittersweet and semisweet chocolates are both plain/dark chocolates. As the names suggest, semisweet chocolate tends to be sweeter than bittersweet - so has a higher sugar content and lower level of cocoa solids than bittersweet. For baking we suggest using bittersweet chocolate which should have 62% minimum cocoa solids and try to use a high end brand such as Ghiradelli. Some British and Swiss brands can be found in gourmet stores which will have 70% minimum cocoa solids.