It is a bit confusing isn�t it? Basically this is about the evolution of secondary schools since WW2. Grammar schools were one of 3 types of schooling introduced post-war by the Government and they were the one for the brightest kids (the other 2 were secondary modern and technical schools). 1960�s governments said that was too elitist, so introduced Comprehensives with the intention of giving every child the same access to good quality education. This was supposed to be the end of Grammar Schools, but actually loads of them continued to exist (some charge fees) and they are probably even more elitist now, ironically.
Academies are the current Labour Government�s version. The main difference is that they are independent of local authorities, although they still get their money from the public purse. The idea is that they are allowed to run more as a business and not get stuck in public sector bureaucracy that many schools have said is causing failure in our education system.
Hope this makes sense and helps.