What is the cause of this barbarous conflict?  What are the solutions?  Who is right?

This is internecine warfare.  There is no right side, although the Israelis have long been able to do as they pleased militarily.  Some Palestinians and other Muslims respond with terrorism, because that is their only option for avoiding a peaceful end to the brutality.  Bear in mind that Hamas only controls the Gaza Strip, and they do support terrorism.

The current crisis appears to have been timed by Israel, and probably provoked by them, despite the very different picture portrayed in the pro-Israeli US media.  Hamas had been sending rockets into Israel in retaliation for an Israeli raid, which may have been a deliberate provocation to get Hamas to respond.  Whether that is true or not, Hamas responded by terrorist rocket attacks.  The attacks were scary, but largely ineffective, causing few Israeli deaths.  The overwhelming retaliation from Israel very quickly caused hundreds of civilian deaths, as the Gaza Strip is one of the most densely populated areas in the world.  Israel has also attacked UN personnel and facilities, which has enraged world opinion.

This little war started during the holiday season in the US, and it takes place when a pro-Israeli lame duck President is leaving.  In other words, this is Israel's last chance to attack Hamas with a relatively free hand.  When Obama takes office, it is unlikely that he will turn a blind eye to such brushfire wars.  Hence, it is no accident that Israel rejected all calls for a ceasefire until the weekend before Obama's inauguration.  They are now saying that they will impose a unilateral ceasefire.  This may well fail, because their stupid act has stirred up a hornet's nest of anger and resentment. 

The solution ought to be UN peacekeeping troops in the Gaza Strip, but I doubt that there will be many countries that will trust putting their troops in as peacekeepers now.  There is not a lot of trust that Israel will honor an agreement not to fire on them, if things heat up again.  In any case, there won't be a possibility for negotiations until after the Israelis have chosen a new leader, probably Netanyahu, who is poorly regarded for his corrupt administration in the past.  Possibly, Ehud Barak, the current Defense Minister, will beat Netanyahu.  As the architect of the current invasion of Gaza, his past reputation as being somewhat dovish will be less troublesome for Israeli militants.  I suspect that Israel will face a much tougher future as a result of this invasion, but that depends on how the other Arab governments in the region feel about it.  Most of them also hate Hamas.