When Shigeru Miyamoto, the creator of Mario (and many of Nintendo's most popular games) was working on Donkey Kong, he wasn't originally looking to build a game around a giant ape and a carpenter with a propensity for jumping (only later did Mario become a plumber). Hoping to develop a game to combat the seemingly-insurmountable popularity of Pac-Man, Miyamoto had originally intended to license existing popular characters that would provide an immediate draw to game-playing audiences. Those plans fell through at the time, and Donkey Kong became the history-making game we know today. However, in 1982, Miyamoto was finally able to acquire the license he had been seeking, and thus Popeye and his friends became video game icons, in addition to comic and cartoon favorites.
Although Popeye isn't as ubiquitous these days as he has been in previous decades (I'll have something to say about the possible reasons for this at the end), if you know anything about Popeye, you can expect at least four elements to almost any Popeye story: the "sailor man" himself, his girlfriend Olive Oyl, his rival Brutus,* and a nearby can of spinach. All of these elements are present in Nintendo's game, as can be seen in this image of the game's opening moments.
The object of the game is to move Popeye around the board collecting various tokens of Olive's affection (hearts in the first round) as they fall toward the water below. Objects at higher levels are worth more points, and if a fallen object stays in the water for too long, it sinks and Popeye loses a turn (it's not good to anger Olive!). Of course, Popeye must do this while avoiding contact with Brutus. Brutus will pummel Popeye into the icy waters if he makes contact, which he will attempt to do not just by chasing him, but also by jumping up to tackle Popeye if the sailor is above him, and by reaching below should Popeye be underneath. If that isn't enough, Brutus can send a volley of bottles flying towards Popeye to knock him out. Popeye can avoid this fate by punching the bottle at just the right moment.
Once per board, Popeye has the option of eating a can of spinach, which generally can be found on the side of the board. To grab a can of spinach, Popeye "punches" it, and he immediately turns red and the famous "Popeye the Sailor Man" song starts to play. The song will play through exactly twice, to indicate how much time Popeye has to use his spinach-powered super-strength to knock Brutus flying into the waters for a temporary cool-down. During this time, Olive's hearts will hover at the height at which they were when the spinach was eaten, and are worth twice as many points as normal.
Another way of stunning Brutus for a few moments is to punch the barrel at the top of the board (only in round one). If you time it just right, it will fall over Brutus' head, and he will be unable to move until it is removed. In theory, Popeye can touch Brutus without harm while Brutus is thus incapacitated, but I don't recommend it. The effect really doesn't last all that long, and it's a pretty senseless risk with absolutely no payoff.
As Popeye collects one of Olive's tokens, it appears at the top of the board. Once Popeye has collected enough of them, he completes the round and moves on to the next one. There are three boards in the game, which repeat in cycle after you have completed them, getting harder to pass each time (and they aren't exactly easy to begin with!).
Other famous Popeye characters make appearances in the game, as well. The Sea Hag appears on the sides of the board (Often two of her at once! Apparently she creates a clone) to throw bottles much as Brutus does, and Wimpy and Swee'Pea appear on the second board, although neither play an active role.
The character of Popeye first appeared in 1929 (just months after Mickey Mouse), and was an enduring favorite for many decades, as the existence of this game attests. He hasn't been quite as popular in years since, possibly at least in part due to changing attitudes about the value of telling children that they can solve their problems by punching out bad guys. I'm certainly sympathetic to concerns that children should not be taught to use violence, but Popeye's had some very positive effects on children, as well. Spinach consumption was reported to have gone up by 33% in a five-year period coinciding with Popeye's animated introduction in the 1930s, and it seems that even as recently as 2010, a study showed that children ate more vegetables after watching old Popeye cartoons (I'd link to the study, but it's apparently one of those things accesible only on an academic database). Surely this a character worth keeping around! Perhaps the recent comic series from IDW (the same company that produces Transformers comics) or the upcoming movie to be directed by Genndy Tartakovsky might help restore Popeye's popularity.
*I'm having trouble finding a definitive source to say that this name, rather than the character's original name of "Bluto," was the one used on the Nintendo video game, but that seems to be the general consensus. For those who don't know, the name "Brutus" was created back in the late '50s when King Features (wrongly) thought that they didn't own the rights to use the "Bluto" name. The character has swapped back-and-forth between names ever since, although some more recent interpretations have suggested that the names refer to twin brothers.