Where is the Promise?
The second letter of Peter is also a "general letter," like 1 Peter. In it Peter addresses all Christians, "those who through the righteousness of our God and Savior Jesus Christ have received a faith as precious as ours" (1:1). This is an even broader audience than 1 Peter, which only addressed Christians in Asia Minor. 
2 Peter addresses a question that no doubt became more and more acute the more time went by after Jesus' resurrection. "Where is this 'coming' he promised?" (2 Pet. 3:4). Peter predicts that there will come a time when scoffers will mock the belief that Jesus is going to return. 
What kinds of things does Peter see them saying? They will say things like, "Ever since the fathers fell asleep, all continues just as it was from the beginning of creation" (3:4, NASB). Since Peter is talking about the future, the fathers he has in mind are not the prophets of the Old Testament but early Christian fathers like Peter himself. He is picturing someone who would say, "Peter and Paul have been dead a long time and yet Jesus hasn't come back yet."
In response, Peter reminds the Church that God's time isn't like human time. "With the Lord a day is like a thousand years, and a thousand years are like a day" (3:8). To that generation, Peter promises that "the Lord is not slow in keeping his promise... Instead he is patient with you, not wanting anyone to perish, but everyone to come to repentance" (3:9). 
2 Peter lives in this tension between reassuring those in the future/condemning the scoffers and bolstering confidence in the original teachings of the apostles and early Christian prophets. 2 Peter 3 addresses the question of Christ's coming and the coming judgment most directly. 2 Peter 2 addresses false teachers most directly. Then 2 Peter 1, the beginning of the letter, starts by bolstering the confidence of Christians everywhere in the original promises and teaching of both Peter himself and Scripture, which in 2 Peter may actually include some of the New Testament writings in addition to the Old Testament!
 Whether you think 2 Peter 3:1 is referring to 1 Peter or some other lost letter by Peter largely depends on your position on the issue mentioned in note 2 below. If you think Peter is the literal author of 2 Peter, then probably the first letter it mentions is some lost letter, since 1 and 2 Peter would then have different audiences. If you think that 2 Peter is a testament giving Peter voice to a later generation, then 3:1 probably does refer to 1 Peter.
 A large number of experts on 2 Peter believe that this letter is a sort of "testament," a literary genre written after someone's death to present what they might say to a later generation if they were still alive. In this scenario, 2 Peter would not be written by Peter but be written several decades later, possibly with audiences knowing it was not really Peter writing. [see, for example, the hypothetical letter of John Wesley that I wrote] In that case, the author would not be talking about the distant future but about what was happening at the very time he was writing. In other words, he would be addressing scoffers in his own day by using the authoritative voice of Peter from the past.
 There is a subtle shift in the tenses here. Peter goes from speaking of what will happen in future "last days" to reassuring his audience in the present.