Yes, the Normans and the Anglo-Saxons are different, as are the Celts from both.
The Normans are an off-shoot of "original" Vikings who settled in the North of France (Normandy), successfully invading England in 1066 under William of Normandy (William I of England).
The Anglo-Saxons are a mixtures of Jutes and Saxons who began to populate Britain in the declining days of the Roman empire. However, they were never part of one united nation (England) until around 850/900, when two most prevalent kingdoms of the time - those of Mercia and Wessex - merged into one, the latter assimilating the former.
The Celts are what you might call the 'indegenous' of the British Isles, their existence here pre-dating the Roman presence and surviving long after it - most notably in Wales and Scotland, the extreme areas to which they were pushed by Roman Imperialism.
Hope that helps, I'm only really aware of their histories from a British perspective, the product of a myopic education system is suppose, but just remember that no race can claim to be truly autochthonous, and every culture has been formed through a myriad of experiences, hence my caution in using the phrase indegenous to describe the Celts.