Ahh. What exactly is an Andalusian? And what the heck is a Spanish horse? What's a "preee"??? or a "Pee Are Eee?"
An Andalusian, Spanish horse, or Lusitano all have certain traits in common - typically grey (sometimes bay or black), they have a straight or convex (rounded) nose, medium ears, triangular eyes, an arched neck, rounded outline, compact body, elevated movements, luxurious mane and tail, and a disposition that is noble and highly trainable.

The Andalusian NAME itself is the common name used in the United States for the horse breed which originates in Spain and/or Portugal. Most people recognize Andalusians as the Spanish version and Lusitanos as the Portuguese version. Some people call both types Andalusians (the generic name). Andalucia is a province in Southern Spain and many of the horses are bred there - hence getting the name Andalusia-ans.
Over there in the "mother country", the horses are known as a type of thoroughbred. Not Thoroughbred in the sense of a racing horse on the track descended from Arabians. But a thoroughbred in the sense of "pure blooded" or "pure bred". This is how it's Spanish name comes about - Pura Raza Espanola (PRE) - or purebred (thoroughbred) Spanish horse. And also the Portuguese name for THEIR horse - Puro Sangue Lusitano (PSL).
Confused?
Don't be.

Just remember PURE Spanish or "PRE" which stands for Pura Raza Espanola and you are looking at a SPANISH horse. Or PURE Portuguese or PSL and you are looking at a PORTUGUESE horse.
Even if a horse isn't PURE Spanish or PURE Portuguese, he may still be recognized as an "Andalusian"! As long as he's either one or the other or a mix of both. The blending of the two lines is known as the Pure Spanish Portuguese (PSP or SP) horse.
So, keep in mind, there are TWO basic breeds abroad - the Andalusian (Spanish horse) and the Lusitano. Minor variations occur between the breeds. The Andalusian was mainly bred as a parade and pleasure horse, a horse for working cattle, and as a king's mount. They are mostly grey with some bays and blacks and the very rare buckskin/palomino. The Lusitano was bred to be a more "athletic" type horse, sometimes having a more Roman head, and come in all colors.
By crossing the two basic breeds, you get an Iberian horse recognized in the United States as a "purebred Andalusian horse" - because both parent lines are purebred.
When it comes to registering your Andalusian or Lusitano - you have several options.
Spain provides an inspection and grading service known as "inscription and revision". These services are only for Pure Spanish horses (Lusitanos and SPs cannot apply). This service is currently overseen by the breeders group known as ANCCE. If your horse passes inspection, you will receive PAPERS FROM SPAIN! known as a "carta" or green card. This process is tricky. Only inspected/approved parents will produce offspring that can be inspected/approved. A horse thus approved is correctly termed a "PRE".
The Foundation for the Pure Spanish horse, a California based group, also provides inspection and grading services. A horse receiving papers from this group will be considered a PRE Mundial (or world PRE). These papers are only for Pure Spanish horses (Lusitanos and SPs cannot apply) and do not transfer over to the true Spanish registry. A horse thus approved is also termed a "PRE" but in reality may not be approved through Spain - which claims to hold the honors of using the term PRE.
The International Andalusian & Lusitano Horse Association issues papers to horses of any pedigree - including Andalusians (Spanish horses) that are not in the Spanish studbook or PRE Mundial as well as issues papers to Lusitanos and Spanish/Portuguese horses. The IALHA does not require an inspection process.
Finally, Portugal also has an inspection program for PSLs.