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.
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.