[Rearranged to be generally chronological. Obviously, butterflies could change all of these significantly, so I'm assuming everything would happen as OTL if it weren't for major changes; as that seems to me the best way to do this.]
9/11: Happens as OTL. Prompts the US-led coalition invasion of Afghanistan
Afghanistan: Invaded in response to the Taliban's harboring of Al-Qaeda, as OTL.
2002 midterms: Democrats gain seats as American's rally to their banner in the aftermath of 9/11.
Hurricane Katrina: Same storm that causes the same damage. Same FEMA that responds, so I imagine the Gore administration is not looked upon favorably. Whether he has a moment similar to Bush's with story time is obviously unknowable.
2004: Gore wins reelection on a thriving economy and progress made in Afgganistan by a decent margin, defeating Republican "heir apparent" John McCain, who (to his credit) put up a good fight.
2006 midterms: Democrats loose seats, as tends to happen in the midterms after the incumbent party wins reelection to the presidency.
2008 Recession: Al Gore's presidency does not shift the foundation of the financial or housing markets, so happens as OTL.
2008 Presidential Election: After 16 Democratic years and an economy that just bottomed out, Republicans win in a land slide against the Democratic sacrificial lamb (especially if said lamb is John Edwards). Jeb Bush seems like as good a guess as any as to who that Republican might be.
Iraq: Not invaded in 2003 as claims of weapons of mass destruction are eventually dismissed. Becomes involved in a mini-cold war arms race with Iran
Iran: Becomes involved with a mini-cold war arms race with Iraq. Each nation is trying to develop an atomic bomb before the other. I couldn't tell you which nation gets the bomb first, but ITTL, we would likely all be expecting an Iran-Iraq war to go hot, this time with nukes to the background of the Arab/Berber Spring (if it hadn't happened already)
The Arab Spring: It was going to happen eventually. The spark might have come earlier or later, who knows? I imagine this would be tied into
North Korea: Same as OTL
the EU debt crisis: Same as OTL
The drug war in Mexico: Same as OTL, unless Gore changes the War on Drugs policy.
Energy: Greater push towards renewables.
EDIT: Didn't notice the "loses Bush in reelection" bit. Frankly, I don't think that could happen. Consider my list one in which Gore wins reelection.