The quick answer is: Because Jesus willfully did penance for 40 days in the desert, we willfully offer up something for the 40 days of Lent (which does not count the Sundays).. We make this offering in remembrance of His offering. The penance we do we use as an opportunity to think upon Jesus' time in the desert - a time of fasting and prayer.
What Do We Give Up?
It can be anything, but should be something we would normally partake in every day. Some give up coffee or soda, others give up meat, or type of meat. Sometimes we give up something which is a healthy choice for us - like giving up chocolate or desserts of any kind. Then, when we would have partaken in that - we think on Christ. It is a way to bring our hearts and minds to Him. Whatever we give up does not have to be severe, it just needs to be something we would miss so that out thoughts can be directed to Him.
Don't Play the Martyr
Others can know what you've given up - but it isn't something you should constantly remind them of. As much as possible, this should be a private sacrifice between you and Jesus.
Forty Days? It Doesn't Add Up!
If you count out all the days from Ash Wednesday to Easter Sunday, it's more than 40 days. and the reason for that is Sundays are "feast days" and thus not intended to be a day of penance so don't count the Sundays and you have 40 days of Lenten observation. This also means that whatever it is you have offered up for Lent does not have to be offered up on Sundays of Lent.
While the Friday penance throughout the rest of the year doesn't HAVE to be abstinence from meat anymore (it still CAN be, and in my humble opinion, SHOULD be), during Lent, as faithful Catholics, we MUST abstain from meat on Ash Wednesday and Good Friday. It must be pointed out, we don't give up meat on Fridays because it is sinful, there is no sin in eating meat, the sin would be the violation of Canon Law - which all faithful Catholics must abide by.