ANSWER: You picked a good question. This 'bruise' has puzzled many a Bible student. These few words in Genesis, which contain the very FIRST Biblical prophecy, are packed with meaning. And, you are right in assuming that the serpent who tempted Eve to sin represents Satan the devil, God's chief adversary.
Just before the verse in question, Adam and Eve had sinned and been confronted by God regarding their disobedience (Genesis 3:1 - 14). The serpent, representing Satan, was the first one to receive punishment from God. The Eternal cursed the serpent to live on his stomach during his entire existence (verse 14). Then he made the statement you are asking about in verse 15.
'I will make you and the woman hate (have enmity between) each other; her offspring (the NKJV and other translations have 'her Seed') and yours (the NKJV and other translations have 'your seed') will always be enemies. Her offspring will crush (Strong's Concordance #H779, the literal meaning of which is 'bruise') your head, and you will bite (the same Hebrew word better translated 'bruise' used a few words earlier) her offspring’s heel.' (Genesis 3:15)
As was stated previously, the serpent represents the devil or Satan. This animal also represent deceived human beings who are, whether they know it or not, serving the purposes of evil ("your seed"). The woman represents Christ and the church ("her Seed"). It has always been a struggle between these two parties, the outcome of which is just staggering. It has to do with God's plan of salvation in producing children for God's family, and Satan endeavoring to damage and destroy as much as possible of that plan. In a small way, it applied to the people of Israel in the Old Testament, (the "church in the wilderness," Acts 7:38) as the Devil enticed the Israelites to sin and disobey.
Most of all it applies to the New Testament church. Satan tried to kill the Messiah, soon after He was born, through Herod. As that failed he used the Pharisees and other enemies to try to make Jesus stumble, right up to the His crucifixion. That having failed, he endeavored to make "her Seed," the church stumble. He has had, tragically, some success there, as some of God's children have lost out and more will. All this is the fulfillment of the phrase (in most translations) that states "you shall bruise his heel."
What about the other part of God's statement, the part where he says that the woman's Seed "shall bruise your head?" Jesus Christ will crush Satan, his power and his empire of demons at the end of this current evil age where man is failing to rule himself (Revelation 11:15). Around five hundred years earlier Daniel prophesied the same event (see Daniel 2:34 - 35, 44).
There are many more prophecies adding more details of this glorious future event that God will bring to pass through Jesus. Tragically, before that fulfillment, mankind will have to learn a very painful lesson. Two world wars have not driven the lesson home yet, but the next one (man's third and last world war) will (Matthew 24:21 - 22).