When a couple has a fight, and one wants to resolve/ solve/ make up, what is the best way to go about it?
Should the person continuously call/ text/ etc. and force the other to talk to them?
Or
Should the person send a text explaining their side, and then wait for the other to reply?

    

I guess the person must try to convince by the best way he can!
By coking food, by saying sorry, by gifting flowers, or anything like that!
Even if that doesn't work he should wait till everything cools down!

Yeah. both of the people need to let the situation calm down and then look at what both of them did wrong. It is important that issues are resolved even if the relationship is not going to continue. There are people that you can go see whom will help resolve these things. These mediators allow both people to have a calm conversation between one another, without taking any sides.

I think that the only way to resolve or make up is for both parties to be in mutual agreement that they want or are ready to do this. Otherwise you are just wasting your time and theirs. If you are not both ready one could just be doing it to get it over with but still hold a grudge against the other person. So all you can do is let them know you are ready to resolve things but if they are not I would not push the issue.

I agree with everybody else. Make sure to calm down before you try to resolve the conflict. Once you both are calm, sit down and talk about your feelings. Don't blame each other. That will just cause more problems.
Sit down in a quiet place and talk about your feelings, and try to work out a solution. Work together, not against each other.
I would definitely stay away from the constant texting. Forcing the other person to talk will not help things. If anything, it'll make things worse.
If you want, send a quick text asking to meet at a coffee shop, home, park, etc. Text something like, " Let's meet at the park during lunch so that we can talk about this and work things out together." Pack a little lunch (bring some flowers too) and go wait for the other person. If she/he doesn't show up, you'll know that this person isn't ready to put the argument behind her/him. Give him/her a few days to think things through. Don't text or call. That just makes you seem needy.
Believe me, when the person is ready and when he/she isn't getting constant messages from you, he/she will be calling you and asking to meet with you.

Yes, definitely refrain from continuously calling or texting your partner immediately following an argument. Men in particular need time and space to be alone before they are able to talk rationally about a conflict. My husband and I struggled with communication a lot in our first few months together, but now we know how to relate to one another.

I think that time can help heal the wounds. Sometimes, the other person simply needs time in order to make up with the other person. They need time to reflect on what happened and to help themselves in the healing process. Just be sure to let them know throughout that period how committed you are to making up with them.