Short answer: Jesus was referring to the authority Peter would have as an elder in making judgments regarding church discipline; he would be an emissary of the divine court, delivering verdicts that had already been determined in heaven.
Matthew 16:19 is an excellent example of why it is crucial to read the text in the original language prior to drawing doctrinal conclusions. This text uses a "future perfect tense" (periphrastic construction), which only occurs six times in the New Testament.1 At this point it would probably be helpful to explain a bit about "tenses" in Greek, and especially the "perfect" tense.
Primer on Greek Tenses
In English "tense" signifies time of action, like "present tense" or "past tense". Greek is very different. In Greek, tense primarily signifies aspect, which basically means kind of action. Now, in the indicative mood (which Matt. 16:19 uses) tense does also signify time, but aspect is always primary.2 (Bear with me.)
There are three aspects used in the Greek tense system: "undefined" / "punctiliar", "continuous", and "perfective."3 Leaving aside the question of time for a moment, let us consider each of these.
"I eat" would be considered "undefined" / "punctiliar" according to the Greek aspect system, as it is unclear whether you are saying something generic, noting reality at a single point in time, describing something continuous, etc.
"I am eating" would be considered "continuous", as it is clear that you are describing an ongoing reality
"You cannot speak to my dog, because I have eaten my dog" would be considered "perfective", because you are doing two things simultaneously: (A) noting something which occurred in the past (you ate your dog), and (B) emphasizing the important present impact of that past action (my dog is presently in the state of having been previously eaten, and thus at present he cannot be consulted.)
To restate this in a slightly different way, the perfective aspect is used to describe something that occurred in the past, but has important implications for the present reality -- relative to the time in focus. You might say "I had an encounter with God" (undefined), "I am being sanctified" (continuous) ... but you might use the perfective aspect to say something like "I have been saved by God Himself!" This would note a past experience, but emphasize your present reality in light of that past experience.
Adding Time: "Future Perfect Tense"
You might have caught my caviot in the prior paragraph that the perfective aspect emphasizes present implications relative to the time in focus. This is where it gets slightly complicated, but it's not impossible to understand if you take your time with it. You can describe a perfective action in the past, present, or future time (in the indicative), which plays out as follows:
"Pluperfect Tense" (Past Perfective): This would be like saying "When my friend visited me this time last year (in the past) I had already eaten my dog, so my friend could not speak to him."
"Perfect Tense" (Present Perfective): This would be like saying "I have eaten my dog, so you cannot (presently) speak to him."
"Future Perfect" Tense (Future Perfective): This would be like saying "If you come to my house this time next year, you will not be able to speak with my dog, because by that (future) time I will have already eaten him."
Grammatical Meaning of Matt. 16:19
What Jesus is saying to Peter is essentially that whatever Peter should "bind" or "loose" (future) will have already been bound / loosed in heaven.
Interpretation of Matthew 16:19
So is Jesus saying that Peter will have "special priest powers"? Not exactly. What He is saying is that Peter will be an effective emissary of heaven in judging cases. To say it another way, what has already been decreed in heaven will be what Peter will subsequently decree.
This is the language of the law court. . . . Jewish people believed that the authority of Heaven stood behind the earthly judges [elders] when they decided cases based on a correct understanding of God's law. . . . by obeying God's law, the earthly court simply ratified the decrees of the heavenly court. In Matthew 18:15-20, Christians who follow the careful procedures of verses 15-17 may be assured that they will act on the authority of God's court when they decide cases. . . . the church by disciplining the person simply recognizes the spiritual reality that is already true in God's sight. -Craig Keener4
Jesus' intention was to give Peter the kind of confidence in deciding cases within the Church that the Jewish elders had in deciding cases in Israel.
1. Mounce, Basics of Biblical Greek, p.235
2. Wallace, Greek Grammar: Beyond the Basics, p.496
3. Mounce, 126-127, 223
4. Mounce, 122