An assault is any offer to harm another person with the apparent intent to commit that harm. It may be a clenched fist, a drawn weapon, or even a threatening gesture. A battery is the unlawful touching of another person. It is not necessary that you harm the other party to constitute a battery. Any unlawful conduct is battery. If one, say, touches a child in an improper manner, even though the child may not be harmed, one is guilty of battery. The primary difference is the assault is the offer or threat to do something to another person; the battery is the unlawful conduct with the other person. Either may stand alone without the other.
The basic difference between these two crimes is that an actual harm must occur in order for battery to have happened whereas an assault can occur even if there is only a threat of harm.
In battery, there must be an actual harm to the victim. The defendant must commit some act that ends up with the victim being harmed.
By contrast, assault can occur even if no harm is done. An assault can occur even if the victim is only scared by some action. In order for an assault to occur, actual harm need not happen. What does have to happen is that the defendant must do something that causes the victim to reasonably fear that they will be hurt.
So, for example, pointing a gun at someone is not battery. However, it is an assault because the person could reasonably feel that some harm would come to them.