Repentance in the context of salvation it either implies faith or is associated with faith. Thus, in a soteriological (salvation) context repentance means to change one's mind about whatever is keeping one from trusting Christ and trust him as the only means of salvation.
In presenting the gospel to people either one on one or in the context of a worship service, I can often sense the Holy Spirit of God drawing them to the Cross of Christ. However, there is also a battle going on within their soul. That battle comes in many different ways as there are people. For some it's not totally understanding this "new" information about their standing before God. For others it could be fear, or some other emotional aspect that's keeping them from trusting Christ. Still others are thinking about what they are currently doing and/or not doing and are not wanting to have to change their lifestyle. Obviously I could go on and on.
However, the root of the problem is self. It's the same problem that Adam and Eve had in the garden - PRIDE. We often here in invitations and when dealing with people one on one that they need to turn from self and turn to Christ (Repentance). The context, I believe, is different for everyone at that moment of responsibility to respond to the gospel. We know that all are born with a sin nature and that we got that nature from Adam. Meaning we are born with a sin deficit. It's like we are born with a bank account where we owe an insurmountable amount of debt because of sin and have no possible means to repay it. Not only that, but because of that sin nature, sinning comes natural. But I'm not done yet because since that is our nature, we then actually do sin and we sin over and over and over again.
Nevertheless, during that moment of whether to trust Christ for salvation or not, most people do not have all that information I just talked about above in their head. However, if the Spirit of God is working through the gospel drawing that person to Christ and convicting that person of sin, whatever that thing or things that is in their mind at that moment repentance needs to take place. But remember, repentance is not just turning away from some sin and a desire to turn over a new leaf and live better. No, it's a turning away from that sin (repentance) and turning to Christ for His forgiveness of your sin and for the imputation of His righteousness.
In doing this, some change their mind (repentance) on concept of God - He is indeed God and Christ is indeed His Son. For other's it's their works and their realization that they can't save you. Admittedly, the seriousness of sin and that you are indeed a sinner in need of a Savior. Thus, once your mind is changed and you place your trust in Him for salvation both the gift of faith and repentance have taken place.