This isn't the most common question, but every so often we'll get a call from a treasurer looking over their broker statements, and finding multiple dates listed for what they originally thought was such a simple buy or sell transaction.
While I don't have a mnemonic for it, here is some information that should make it easier to keep these dates in order when doing your book-keeping

The Record Date is when the club has to actually own, and be holding the stock, in order to receive a dividend. The Ex-dividend date is the date on which any new purchases of the stock will not be on record in time to receive a dividend.
The Ex-dividend date will be no less than two business days before the Record date.

The reason these dates are used is because in the current market, when you Buy or Sell (Trade) a stock, that company's books must be updated (Settled) within three days of the transaction. This is why your broker might show a Trade date, and then a Settlement date on their records.

To give an example of how all these dates work together:
Let's say you know that the Record Date (Date of Record) for a stock your club is interested in is on Thursday of a given week. Assuming the company follows a standard procedure, that means the Ex-dividend date will be no latter than Tuesday.
This means that if your club has been thinking about buying the stock, the broker would need to make sure the transaction (Trade Date) is no latter than Monday, in order to make sure that the Settlement Date comes on Wednesday, BEFORE the Record Date of Thursday.

Again, this can be a bit of a handful to keep track of, but hopefully it gives you a bit more of a handle on what some of the information on the broker statement is trying to show.