In the simplest terms, "hard money" is from political donations that are regulated by law through the Federal Election Commission. "Soft money" is money donated to political parties in a way that leaves the contribution unregulated. The difference boils down to a few crucial words and one administrative ruling.
The largest expense in a modern political campaign is advertising, and most of the controversy about hard or soft money ultimately points to how much advertising a candidate can buy. When we think about political advertising, we tend to think about an ad that may say many things, but finally says, "Vote for Candidate X." If an ad tells you, in plain terms, which candidate to vote for, then it's a political ad and must be paid for out of the candidate's campaign funds, or out of special funds of the political party. The contributions to these funds are regulated by law: Corporations and labor unions can't contribute to campaigns for federal elections, and individuals are limited to contributing a maximum of $1,000 to a federal candidate, and $20,000 a year to a political party for the purpose of telling people whom to vote for. These rules are long-standing (some going back to 1907), heavily policed and well known. In 1978, they also became much less relevant.
In 1978, the Federal Election Commission issued an administrative ruling that the funding rules established by law only applied to political campaigns, and not to "party building" activities. The commission didn't go into great detail about what constituted a party building activity, basically defining it as something that didn't explicitly tell people to vote for a specific candidate. The ruling was issued, and political parties uniformly ignored it until 1988. In the 1988 presidential campaigns, people working for both major parties discovered the "loophole" created by the 1978 ruling, and the race for soft money was on.
Because soft money is not regulated by election laws, companies, unions and individuals may give donations in any amount to a political party for the purpose of "party building." Party building may include ads that educate voters about issues, as long as the ads don't take the crucial step of telling voters which candidates to vote for. For example:
Candidate X runs an ad that says, "I am a good person. Candidate Y is a bad person. Vote for me on election day." Because of the "Vote for me..." portion, this is a political ad, which must be paid for with "hard money."
Candidate Y runs an ad that says, "Candidate X has a record that includes awful things. If these awful things continue, people will come to your house, steal your money and shoot your dog. Be sure to vote on election day." Because the ad "educates" people on an issue and doesn't tell them to vote for a particular candidate, it's party building, and can be paid for by soft money.
Here are some interesting links: