Are all plants photosynthetic? What about those plants that eat insects? Are they photosynthetic then? If they are not, why are they called plants?

No, not all plants are photosynthetic - I know of one family (Orobanchaceae, the broomrapes) which contains several genera of entirely parasitic, non-photosynthetic plants. Most of the insectivorous plants are photosynthetic - that's why they're green.  The insect-eating is generally a way of gaining essential nutrients from the low-nutrient environments they live in.

As for whether they're still plants, it's important to remember that, in phylogenomics, children keep the family name even if they break tradition. Non-photosynthetic plants are still members of the plant kingdom because they are descended from the ancestral plant specimen, and so are still classed as plants, in the same way that flies without wings are still classed as Diptera.

Richard is correct that there are a range of parasitic species that are not photosynthetic, and some don't even produce leaves.  Photosynthesis is, after all, the process by which light energy is harvested and stored as chemical energy, and leaves are the most usual way that plants expose lots of tissue to light in order to house the photosynthetic process.

Equally important, though, is to realise that plants are not the only organisms which photosynthesise - a number of different photosynthetic pathways have evolved, and not all utilise the same wavelengths of light.  So outside of plants there are a number of different photoautotrophic (= organisms that feed themselves using light energy) groups, including some bacteria and several groups of Protoctista, including the so-called red algae, brown algae and green algae.