Kaddish is often described as the Jewish prayer for the dead because it is recited by mourners on behalf of their deceased relatives, but this is, in fact, only one of the ways in which the prayer is used. The prayer itself makes no mention whatsoever of death or that someone has died. Rather, it is a prayer which praises and extolls God and looks forward to the Messianic age. A translation of the prayer is set out below.
The Kaddish prayer is in Aramaic, not Hebrew, except for the final sentence. This is because it stems from a time in history (the period of the Second Temple in Jerusalem, 516BCE - 70CE) when many Jews used Aramaic as their everyday language rather than Hebrew. To the untrained ear it can be difficult to distinguish the two languages as they are related and share common features, including much vocabulary, but they are nevertheless distinct languages and each has a separate system of grammar, including verb conjugations.
The name of the Kaddish prayer indicates holiness and reflects that the prayer sancitifies God by publicly affirming His goodness and holiness. As the core of the prayer is a public affirmation of God's goodness, it is said only in the presence of a minyan (prayer quorum) and not by a person who is praying alone.
Originally Kaddish was said by rabbis when they completed giving a sermon. Then it was adopted and said by congregations when they finished a section of a prayer service, hence the current practice of Kaddish being said a number of times during a prayer service, and especially after major prayers.
Later again, although there is uncertaintly as to exactly when, it began to be said by mourners on behalf of their deceased relatives in order to accrue religious merit to them. The underlying idea is that it reflects well on the deceased if they brought up their child to be able to stand up and publicly affirm the goodness of God even in the face of the overwhelming personal loss of a parent. However, a countervailing influence was the Jewish belief that the soul of even the most evil person spends no more than one year in Gehenna (the Jewish version of hell) before being released to go to Olam Ha'ba (the 'World to Come'). Out of concern that saying Kaddish on behalf of a deceased parent every day for 12 months after their death would imply the parent was a very wicked person expected to spend the maximum time in Gehenna, the practice developed of reducing the time frame by one month and saying Kaddish on behalf of a deceased parent for 11 months after their death. Kaddish is then said again on behalf of a parent on every yahrzeit (anniversary of their death).
There are a number of versions of the Kaddish prayer. Best known is the Mourner's Kaddish (Kaddish Yatom), but there are also versions known as Full Kaddish (Kadesh Shalem), Half Kaddish (Chetzi Kaddish), and Scholar's Kaddish (Kaddish d'Rabbanan), and there are guidelines as to which version of Kaddish is said on which occasion.
May His great name be exalted and sanctified
in the world which He created according to His will.
May He establish His kingdom
and may His salvation blossom and His annointed [the Messiah] come soon
during your lifetime and during your days
and during the lifetimes of all the House of Israel,
speedily and soon. And let us say, Amen.
May His great name be blessed forever and for all eternity!
Blessed and praised, glorified and exalted,
extolled and honoured, adored and lauded,
be the name of the Holy One, blessed be He,
above and beyond all blessings,
that are uttered in the world. And let us say, Amen.
[Half Kaddish ends here; Full Kaddish continues:]
May the prayers and supplications of all the House of Israel
be accepted by their Father Who is in heaven. And let us say, Amen.
[The following section is said only in Kaddish d'Rabbanan, Scholar's Kaddish:]
For Israel, for the Rabbis and their disciples, for the disciples of their disciples,
and for all those who engage in the study of Torah
in this (holy) place or in any other place,
may there be abundant peace, grace, loving kindness, compassion, long life,
ample sustenance and salvation from the Father Who is in heaven (and earth).
And let us say, Amen.
[All versions except Half Kaddish continue:]
May there be abundant peace from heaven and good life, satisfaction, help, comfort,
refuge, healing, redemption, forgiveness, atonement, relief and salvation
for us and for all His people Israel. And let us say, Amen.
May He Who makes peace in His high places
grant (in His mercy) peace for us and for all his people Israel.
And let us say, Amen.
Judaism 101: Kaddish
Ohr Somayach: Insights into Saying Kaddish
Virtual Jewish Library: Mourner's Kaddish
How to Say the Mourner's Kaddish on You Tube - video with introductory explanation by Chief Rabbi Jonathan Sacks, then detailed instructions on saying the Kaddish accompanied by a transliterated text
Mourner's Kaddish on You Tube (a slow and clear reading of the Mourner's Kaddish)
The Kaddish on You Tube (video of Kaddish being chanted using the tune traditionally used in synagogues)