All indications are that many European airports will be opening, sometimes with restrictions, tomorrow. Of course, the situation is fluid, and if the volcano starts sending ash Europe-wards in the next few days or weeks, then we could have a replay. (Update: A new ash cloud is heading toward the UK, according to SkyNews)
The bad news is that literally thousands of people have been stranded outside their home countries, and these folks will get priority from the airlines for whatever seats are available. So if you're merely starting a vacation or business trip, you may not get where you're going anytime soon. The good news is that airlines have provided full refunds, even on non-refundable fares, and many people have simply decided to cancel their trips or postpone them, which will free up seats. If you do not wish to travel, do not cancel your reservation on your own; wait for the airline to cancel your flight. Otherwise you may incur a change penalty. Most airlines are allowing passengers to make one schedule change for free.
The bad news is that airlines have slashed capacity between Europe and North America, and there are few seats to play around with.
Most British airports will open tomorrow (Tuesday) as will most French airports (Air France calls this a "partial opening"). Virgin Atlantic is still cancelling all flights out of London Gatwick tomorrow, and is not accepting any new bookings for travel up to and including April 30, 2010. It's likely that other airlines will also not be accepting new bookings for the next few days or weeks. KLM operated several flights today, and plans to fly eight more tomorrow.
British Airways and other carriers are requesting financial aid from their governments and the EU.
When the same Iceland volcano erupted in the early 1800s, it was active off and on for 14 months. Let's hope history doesn't repeat itself.