In ascending order
Assault - A puts B in fear of immediate force against them
Battery - A applies unlawful force e.g. by slapping or punching B
This is a rather technical distinction and both legal "common assault", and colloquially what we call "an assault", much more closely match the definition for battery*
Assault and battery come from common law definitions, not the Act itself*. However, as they are offences against the person, they are involved in the framework.
Assault occassioning ABH (s.47 Offences Against the Person Act 1861) - "ABH" - degree of injury caused that is "more than transient and trifling". Scratches or minor bruising are more likely to be common assault, fractures and bruising ABH.
Grievous bodily harm with intent (s18 OAPA 1861) and without intent (s.20*) are the most serious non-fatal offences, for example which might lead to hospitalisation but the victim would not be killed. The maximum sentence is life imprisonment.
How was this explanation? Once you have got to grips with these basic descriptions, you will be able to add detail, including case law.
If this explanation helped, feel free to arrange a free "Meet the Tutor" meeting and I will be happy to help you further. I have just graduated with a First in Law from the University of Exeter.