Specifically that which is unsubtle or inauthentic in its way of trying to elicit a certain response from a viewer, listener, audience, etc. Celine Dion is cheesy because her lyrics, timbre, key changes, and swelling orchestral accompaniment telegraph 'i want you to be moved' instead of moving you. Gold chains on an exposed hairy chest are cheesy because they shout out: "I have money and I am manly" instead of impressing a woman in a more subtle way, or allowing a woman to form her own judgments. The excessive showing off suggests he's compensating for what he does not have--i.e., he's actually poor, insecure, or short with an inferiority complex. Cliches are often cheesy because they are an obvious and artless way of making a point. A movie might be cheesy if it contains 'on the nose' dialogue, like "I can't live without you" or "You had me at hello."
Jersey Shore is cheesy because they are trying too hard to be 'cool' while it's just bunch of crap.
The most significant use of the term was in 2002 during a series of legendary foosball games at the University of Northern British Columbia a player, having lost a number of games began to shake the table. In his inglorious method of distracting opponents, he began to score goals, even winning some games.
The entire universtiy foosball population began calling his goals cheesy, although the reason remains unclear, especially seeing as he didn't even go to the damn school. Some refer to the goals as cheesy because they "Stink". Other use the term because the player that scored such a fluke of a goal operated a cheese factory on his parents farm. Still others probably came up with it while blazing a fattie.
To this day, poor goals in foosball are known as cheesy, and the term has spread as far as Europe where the French say "Fromage" in the event of poor goal.
The word cheesy is often defined simply as "sentimental", but there is a key distinction between the two. A situation is cheesy when the characters, their motives, and/or their emotions are not believable. It is a sentimental moment (it could be extremely sad or happy), but there is a weak connection to the characters or their situation, thus making it cheesy.
Many people confuse the two, and it is a matter of opinion to a certain degree, but things can be sentimental, emotional, or dramatic without being cheesy.