I received a great question this last week about baptism and about what exactly is meant by "promises" for children as they are baptized. 

Ordinarily, in a baptism, the parents answer three questions, the second of which asks, "Do you claim Gods covenant promises in his (or her) behalf, and do you look in faith to the Lord Jesus Christ for (his or her) salvation, as you do for your own?"

In this question, we confess that God has "covenant promises" for children. What does it mean?
The idea that covenant children have a promise from God comes from Acts 2:39, 
Acts 2:37-39 (ESV)
37 Now when they heard this they were cut to the heart, and said to Peter and the rest of the apostles, Brothers, what shall we do? 38 And Peter said to them, Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit. 39 For the promise is for you and for your children and for all who are far off, everyone whom the Lord our God calls to himself.

Verse 39 says that the promise is for your children. What does that mean, and what does it not mean? Verse 39 is speaking about the gift of the Holy Spirit (verse 38), a reference to all the saving works of God, including all His covenant promises.
If you look at the language of Acts 2:37-39, you will see that it uses the Old Testament covenantal formula of the promise being "for you and your children". This is covenantal language that is applied to both the New Testament church and the Old Testament people of to Israel (see Genesis 17:7 for example). Peter adds the comment "for all who are far off" to show that even the dispersed Jewish people continue to carry Gods covenantal promises wherever they go. 
In this we have a clear affirmation that God bestows promises on the children of believers. These covenant promises are both unilateral and conditional. They are conditional, not on the condition of works, but on the condition of faith. They are unilateral in the sense that God gives his elect the faith they need to walk in those covenant promises (Ephesians 2:8-9).
As we speak about our children, we are comfortable to say that God makes covenantal promises to them promises to save them from sin through Jesus Christ. What do we mean by a "promise"? Remember, a promise is not a guarantee. I like to use this illustration: say I promise we will go out to get ice cream. I may make a promise, but there are times that the promise may not be fulfilled, even though it is a promise. We might have an accident, or my children may act like total jerks and I realize we need a quiet night at home or an early bed time (It would have to be pretty bad). Promises can contain some sort of condition to them. The promise of God for salvation is conditioned on their personal faith (Romans 10:9-10). 
A guarantee is a bit different. A guarantee is unconditional. If I guarantee ice cream then nothing should stop that, even bad behavior or accidents. (I have learned that I personally cannot make many guarantees in life.)
The same may be said about a marriage vow. A bride may vow "till death do us part" but after years of suffering under a husbands adultery, she may recognize that the conditions of her wedding day promise have been so violated that she may biblically exit the marriage (just as Jesus permits in Matthew 19). A husband or a wife doesn't guarantee, they will stay married, no matter what. Instead, they promises to uphold the commitments of their marriage vows, and combined with the spouse fulfilling their own marriage vows, the marriage will succeed. We can make marriage promises, not marriage guarantees.
So we believe God makes promises of salvation for our children, but not a guarantee of salvation. God promises them salvation through faith. But they need faith. Will they put their faith in Jesus Christ? If in His sovereign mercy he gives the gift of faith, we believe they will believe, and we believe they will enjoy the promise of salvation.
A sacrament is a sign and seal of the covenant of grace. Baptism signifies the forgiveness of sin for Gods people. Baptism is sealed with Gods authoritative declaration, for anyone who believes and particularly for that child when they believe.
What that means in this case the child carries the sign of promise, and they need to receive the substance of that promise by receiving Jesus Christ by faith. As parents (and as a church), we have special hope to believe that, as they grow in the Christian home, they will also believe in the Lord Jesus Christ. Do we have a guarantee they will? No. So we keep praying for them and calling them to believe the promise of God, that salvation comes to those who believe in Jesus Christ by faith. But we have a hope and an expectation that, in time, they will.
Practically speaking, this means we disciple children in the Lord from the earliest ages. We teach the fear and admonition of the Lord. Part of that discipleship is to make their own confession of faith in front of the church. We disciple them as covenant children, we do not treat them like as atheists or unbelievers from the start, we treat them as covenant children and teach them to walk with the Lord, and to make their calling and election sure through the obedience of faith and their own confession of faith.

We remind them of the promise they carry by their baptism, that they are numbered among Gods covenantal people. As they carry the sign of baptism they should repent and believe.
It makes me ask, what about an unbaptized child? They also have the same covenant promises. We have the same hope for their salvation and the confidence that God will use our Christian home for their salvation. Without baptism, they miss the sign and seal that God uses to testify to faith, to give assurance, and even use to bring people to faith. But the promises are still there, even if the sign and seal is missing.
The covenantal promise is that all who believe in the Lord shall be saved. While we as parents have special hope God will save our kids, in baptism our children carry the promise of God in a physical way. Parents need to be sure they teach those promises to their children and the children need to respond to that promise by faith. It is a call to follow Christ with an assurance to all who do it.

God must give our children faith if they are to be saved (Ephesians 2:8-9), and that it doesn't come from parents, manipulation or from works. It reminds me of John 1:13. There is no justification for passivity here. As the parents make a number of vows they also promise to teach the word of God to the children because faith comes by hearing and hearing by the Word of God. Without helping our children understand God's Word, we shouldn't be surprised if they don't believe, and miss out on the realized promises of God.
The covenant promises of God are (in one fashion or another) recognized by all believers. What is not always recognized is the implication of those promises, and what is not always agreed upon is whether those promises warrant the baptism of a child/infant. But the particular view of baptism is much smaller than our confessed trust in God for the salvation of our children.