Long codes are standard phone numbers like +1 (407) 555-0100 or +44 7700 900396. Short codes are the much smaller, 5-6 digit numbers you typically see for promotions/marketing, like "Text A to 55444 to vote for Billy McSingerguy".
Long codes are available for instant addition to your Tropo application, just pick a country and choose an SMS-enabled number. Short codes require an approval process with the mobile that is measured in weeks or months.
Short codes can send more messages faster than long codes. Short codes are country-specific (meaning a person in the UK can't text a US short code).