Earnest money, also called "good faith" money is money you put down to indicate that you are serious about making an offer on a home. Basically it stops 500 people from just "making an offer" on every house out there. Not just everyone will be willing to write a check for a few thousand dollars on every home out there. This shows the seller that you are interested in their house.
It also can make a statement about how strong your interest is in the house. Typically, a buyer will write a check for 1% to 2% of the sales price of the house. If you are very serious, or are in multiple offers, you may want to increase that to the 3-5% range. Writing a $10,000 check for a $200,000 house is a strong statement!
TrustFunds is a way to transfer earnest money funds from you (the buyer) to the listing broker without the risk of paper checks. Some brokers may not accept TrustFunds and in that case, we will still have to use the paper check route. Paper checks are less secure for a few reasons. Paper checks are sometimes lost in the paperwork; they are sometimes forgotten on a copy machine, they are scanned and emailed (which itself isn't very secure), they have account numbers on them that other parties do not need to have. For these reasons, I believe that TrustFunds is a better way as it blocks more people from receiving your account number.