A fluorescent lamp is filled with mercury (Hg) gas. When you switch on the lamp the gas starts emitting light. However in contrast to the sodium lamps which are widely used in street lighting (the orange lamps), mercury emits UV light (mainly 254 nm) which is not only invisible for us, but could also harm us in the same way UV light from the sun can harm us.
To overcome this problem scientists use phosphors. These phosphors are insulating materials to which impurities are added. These impurities absorb UV or blue light (depending on the material and the application) and reemit visible light (e.g. green or red). If you take a closer look at the lamp you will be able to see these phosphors since they are applied as a coating on the inside of the lamp and are responsible for the white color of the glass.
The most widely used phosphors contain lanthanides as impurities. In the case of your spectrum the red lines come from trivalent Europium whereas the green emission comes from trivalent Terbium. However on the above wikipedia link you will find a huge list of used phosphors containing also transition metal ions like Manganese and various others.
A final note which is not really related to your question but it is closely related to my research and nice to know: the glow in the dark materials work in a similar way, they only have the possibility to store the absorbed energy before reemitting it again.