Odysseus' wife has suitors because they all want the wealth and power that comes with being married to her. With the absence of Odysseus, and his assumed death, his wife is subject to having to choose a man to be her husband and Ruler of Ithica. A woman in that time had very little power and since she was very wealth all the suitors wanted a chance to have her money and wealth. On another note, she was also very pretty woman and so was easy on the eyes.
Odysseus' wife Penelope has suitors because of Odysseus' long absence. They assume that he is dead, and they hope to marry Penelope to inherit all that she has.
The suitors believe Odysseus to be dead. They wish to inherit his vast fortune and kingdom. They don't wait to begin sampling some of his delights, holding lavish parties and refusing to leave until she selects one of them to marry.
Penelope will not be convinced without proof that her beloved has indeed died. She tells the suitors that she cannot marry until she finished a tapestry. Each night she cleverly undoes the weaving so it is never completed.
Upon Odysseus return, the suitors pay for their treachery. Not one escapes alive.