In these roles, players try to cooperatively solve a series of blue, yellow and red alerts before failing five such alerts, or getting the Enterprise destroyed. Crew members have different abilities, and these abilities can help the team solve alerts — which is how players score points in the game. Both the crew and the alert encounters differ depending on the era in which the game takes place. Whichever crew you're a part of, can you score twelve points before encountering failure or destruction?
Star Trek: Five-Year Mission includes eight double-sided character cards, and each player takes one at the start of play. Identical roles are paired together, so Picard and Kirk are back-to-back, as are Spock and Riker, etc. Only one captain per mission! (I don't know whether you can mix-and-match characters from the original series and Next Gen, but Mayfair can't police what you do at home, so have at it.)
Ideally I'll state all the details of play correctly, but I know from long experience that con demos can deviate from the published game experience. With that caveat out of the way:
Dice come in three colors, and each player starts with five dice. Players take turns in clockwise order, and to start a turn you reveal an alert card for that round. In general, blue alerts are easiest to solve, then yellow alerts, with red alerts being the most difficult — but also the most likely to be worth points when you solve them. When the game starts, you can reveal an alert color of your choice, but sometimes revealing a blue alert will force you to reveal a yellow one (as in the "Beam Up" blue alert in the image below), or a yellow one will force you to reveal a red. If you have more than three alerts in a color, then the bottommost alert counts as an automatic failure.
Once you reveal the alert, you refill your supply to five dice (unless you have dice locked due to injuries), then roll those dice up to three times, after which you can attempt to resolve alerts. If a die on a card is colored, then the die must be that color; if the die is white, any color die can be used. When you place a die, it must either exactly match the number shown or (if an arrow is present on the icon) be higher or lower than the number depicted.
Some cards have a dark border around some or all of the dice shown on that alert, as with "Fire Torpedoes!" and "Romulan Warbird" in the image below. In this case, you must place all of the dice within this border on the same turn. (Someone has started to resolve "Fire Torpedoes!", but they either didn't have a pair of 2s or chose to place those dice elsewhere.) Dice remain on an alert card until it's resolved or failed.
Some alert cards have special effects that take place when they come into play or when they're resolved or failed. "Tribbles", for example, cause one damage to the Enterprise when it's revealed, and as the Enterprise takes damage, you're restricted in the colors of alerts that you can reveal. Take a couple of points, and you can no longer choose to reveal blue alerts; take more, and you can now reveal only red alerts, which seems like a quick slide toward doom. You can spend dice to resolve damage to the Enterprise, but the more damaged the ship, the higher the number you need to resolve that damage.
Alert cards sometimes have effects that mess with the co-operative nature of the game. "Alien Contest" has you place a timer on the card, and when the sand in the timer runs out, you've automatically failed that alert. Zoom! Time to start rushing your turns, which means you'll possibly get sloppy and overlook possible plays or special powers (as some resolved alerts grant you a one-time power that you can hold in reserve). Another alert might break the Enterprise's com link, which in game terms means that you can no longer speak to one another for as long as that alert remains in play. No more supersized dice-swapping for Wesley...
I thought that we were doing well during our game, making progress toward the twelve VP victory condition at a fair clip and putting our special powers into play in sensible ways, but then we started bogging down and blowing one alert after another. Halfway through the game I completely forgot about the ability to fix damage to the Enterprise, and I think that others did, too. You're trying to see everything at once, but it's a lot to take in, which is probably where the floor of three players comes in on the player count. Yes, you could play this on your own with any number of character cards in front of you, rotating through those cards with each turn, but that would definitely lessen the "we're all in this together, team" feeling that this design has tried to create.
In conclusion, "Shut up, Wesley!"