On the main page of the wiki, you can find the core rules of play. While these are very detailed, they were actually designed so a GM with a basic understanding of them could run the game using that basic knowledge without having to refer to the manual. Feel free to peruse these detailed rules through the links at the bottom of this page. However, this page will outline the very basics.
Percentile System
The Hero at Heart core rules are based on a percentile system. Thus, if a character has an Intellect of 45 and a Knowledge skill with one rank in it (giving them a +5 bonus when attempting to perform actions supported by that skill), it means they have a 50% chance of succeeding in that action. This is further broken down into Success Ranks, which help the GM figure out exactly how successful the action was. Ten percent of the time a character succeeds, a character will do either extremely well or extremely poorly at an action. Forty percent of the time, they will do fair or good. Percent scores can be raised above 100, which means a character with a 150 as their target number will be successful almost all of the time, with a very slight chance of failure (if they roll a 100, it is an automatic failure regardless of their score, because there is always a chance for failure). However, that character has a 50% chance of rolling an Excellent or Good success.
Rolling
All rolls are made on percentile dice (1d100 or 2d10), which means the roll will be between the numbers 1 and 100. The GM decides what number they will be rolling on, based on their attributes, skills, and characteristics. He adds in bonuses or penalties for any outside forces which may be acting on them (he can either add a bonus or penalty, which changes the target number they need to roll on, or he can choose to give them bonuses or penalties to their success rank, such as by saying that a character who rolls a Fair success may have a +1 rank bonus, giving them a Good success). The player then rolls, and if they get a number equal to or less than their target number, they succeed. For example, a character with an Intellect of 45 and one rank in a Knowledge skill must roll on a target number of 50 (45 plus the bonus of 5 points they get from the one rank in their skill). The GM may decide that the knowledge is not very widely-known, and so incorporates a penalty of -10, meaning that their target number is now 40. The player rolls, and if they get a 40 or less, they succeed. Any higher than that, and they fail.
Success Ranks
The roll may have different levels of success to it. See the section on Rolling and Success Rate to find out how to calculate this. The ranks of success are Poor, Fair, Good, Excellent, Failure, and Critical Failure. The GM should decide how successful an action was based on these scores. For example, using the previous idea of a Knowledge roll, the GM may decide on a Poor that the character has some hazy memory of the knowledge he's trying to call up, but only enough to be not very helpful. On a Fair, the GM might tell him something useful to know, but make it cryptic enough to where the character doesn't have all the information they need without digging deeper. On a Good success, the GM should tell the character everything he needs to know, and on an Excellent he may even give some extra, helpful knowledge. On a Failure, the character obviously has no idea about the subject at hand, while on a Critical Failure the GM may give the character misinformation and tell the player that they have to act as though they believe it to be true.
There can be ranks higher than Excellent, and the GM can scale the success of the action accordingly.
Opposing or Assisting Rolls
The GM may allow rolls that help other rolls. It is best to think of the success ratings as numbers... Poor is -1 (below average), Fair is 0 (average), Good is +1 (above average), and Excellent is +2 (well above average). Any additional ranks beyond this are +3, +4, etc..
If an action takes place that helps or hinders another roll, the success rating is applied as a bonus or penalty to the success rating of the roll. Thus, if one character attacks and rolls a Good, but the target tried to dodge and rolled an Excellent, the success of the attacker would be reduced by two ranks (because the target gets a bonus of +2, so the attacker would get the opposite of that). Thus, the attacker's Good result would be reduced to a Poor. If two characters are trying to call up information together, and the first rolls a Poor (-1), the second may roll an Excellent, but because the other character's penalty hindered them, their Excellent would be reduced by one rank and become Good.