The details are at their page here, but here’s the gist:
Text REDCROSS to 90999
(I’ve just tried it myself, works)
from their FAQs
Does the full $10 go to the Red Cross?
Your text donation will support will support our disaster relief efforts to help those affected by the earthquake in Japan and tsunami throughout the Pacific. On those rare occasions when donations exceed American Red Cross expenses for a specific disaster, contributions are used to prepare for and serve victims of other disasters. mGive, the agency working to collect and process donations on behalf of the Red Cross, charges a $.25 fee per donation.
Will my wireless carrier charge me for text messaging to 90999?
When you text the word REDCROSS to 90999 you donate $10 to support our disaster relief efforts to help those affected by the earthquake in Japan and tsunami throughout the Pacific. A onetime $10 donation will be added to your phone bill. Depending on your carrier agreement, message and data rates may apply. Your $10 donation will appear on your regular monthly phone bill.
How many times can someone make a text donation via 90999?
Most wireless carriers will allow you to donate up to $30 per month to a specific keyword (like REDCROSS). If you wish to donate more than $30, please visit redcross.org or call 1-800-RED CROSS.
How will people know that this texting program is legitimate?
To document your donation, after you send a text message you will receive a confirmation text that asks you to reply “YES” if you intend to give a $10 donation to support the Red Cross. A thank you text will follow.