Mono is a disease that manifests itself in a fever and swollen glands, as Fluent so adequately described. It causes people to be very tired and very sick and achey, and can often last for months.
Billy used mono as an excuse to hide Jacob from Bella because it is the perfect excuse for someone who needs to be hidden from view for awhile. Mono often affects teenagers (personally, my sister had it when she was a teen!), which Jacob is, of course. Mono also starts with a high fever, which fits Jacob's heat flash when Bella last saw him. It also makes people very sick and very exhausted for a long time. If someone had mono, they could very well be sick for awhile and not be able to have any visitors. Billy came up with this excuse in order to hide Jacob's lycanthropy (werewolf-ism) from Bella. If Billy had just said Jacob had the flu that was going around, he would have been over it in 24 hours or so, and Billy was really trying to buy some time for Jacob to adjust to being a werewolf, and also to protect Bella from being around him, which would place her in danger. He also wanted to protect the secret of the werewolves from Bella, who might find out.
Mono is an acute disease characterized by fever and swollen lymph nodes and an abnormal increase of mononuclear leucocytes or monocytes in the bloodstream. It's not highly contagious but some believe it can be transmitted by kissing.
In the film, mono is the supposed reason why Jacob Black isn't allowed to make contact with Bella. It's also pretty suggestive of what Jacob's father assumes exists between his son and Miss Swan (re: it's contagious nature via kissing)
Unfortunately, that's all I can give. Very few medical sites provide a dumb answer for us to understand, but I'm guessing that it's used as the excuse for Jacob's inability to meet Bella not only because of its contagious nature but because it's symptoms must involve a buff Taylor Lautner (reference to its tell-tale signs of swelling and fever etc, which would explain his body and the hot bursts that accompany his transformation into a wolf).