23 If then you are bringing your gift to the alter and should remember that your brother has something against you,
24 you must leave your gift there before the altar, and firstly be reconciled to your brother. Then come and offer your gift.
When Jesus spoke these words concerning reconciliation He was not telling the Jews something new. The Jews knew that for their sacrifice to be effective they had to seek to be reconciled with any brother who held something against them.
Jesus was reminding them about this.
The following is what is written in the Law of Moses; it is quite lengthy, but it will give us some understanding as to what Jesus was referring.
[6:1] And the LORD said to Moses,
 “Suppose some of the people sin against the LORD by falsely telling their neighbour that an item entrusted to their safekeeping has been lost or stolen. Or suppose they have been dishonest with regard to a security deposit, or they have taken something by theft or extortion.
 Or suppose they find a lost item and lie about it, or they deny something while under oath, or they commit any other similar sin.
 If they have sinned in any of these ways and are guilty, they must give back whatever they have taken by theft or extortion, whether a security deposit, or property entrusted to them, or a lost object that they claimed as their own,
 or anything gained by swearing falsely. When they realize their guilt, they must restore the principal amount plus a penalty of 20 percent to the person they have harmed.
 They must then bring a guilt offering to the priest, who will present it before the LORD. This offering must be a ram with no physical defects or the animal’s equivalent value in silver.
 The priest will then make atonement for them before the LORD, and they will be forgiven” – Leviticus 6.1-7 (NLT).
It would appear that Jesus knew that this requirement of the Law was not being honoured, and He was telling the people that in order for their relationships to be restored with God they must firstly seek reconciliation and restitution with their brothers.
In Jesus we are reconciled to God, our Father, and He is our restitution, since He had paid the price of our sin.