Come to the disABILITIES Board and we can help you out.
As Tissa mentioned, the card is called a Guest Assistance Card. We abbreviate it as GAC on the boards here so we don't have to type so much.
The GAC is not intended to (and often won't) shorten, eliminate your wait in line or let you go to the head of the line. What it is meant to be is a tool for CMs so they can tell easily what assistance is needed, particularly for people with invisible disabilities. Some of the things it might allow would be a quieter place to wait, being allowed to bring a stroller into line and have it treated like a wheelchair (this is something that a lot of people with children with autism find helpful - you can bring a little "safe place" into the line with you).
The GAC is issued to the person with a disability; they need to be with you when you request one and when you use it. It is usually issued for up to 6 people (5 plus the person with a disability). In some cases, you will be asked to split into smaller groups; that's usually in situations where the waiting area or the seating area for people with special needs are small.
And Tigger & Belle is right; if you go during off season, the lines are pretty much a non-issue. Using Fastpass is also a good idea. You can send someone in your party ahead to get fastpasses; that way, your whole party (no matter how big) can stay together. You put your park passes into the fastpass machine and get a Fastpass with a return time. When you come back during your return time, your wait in line will be less than 15 minutes. One advantage to using Fastpass is that you know what will happen; you get there, you show your Fastpass, you have a short wait in line. Using a GAC (even on the same line at a different time of day), you may not always experience the same thing - where you wait, how long or even being given a time to come back, depends on ride factors that you don't control (like staffing, how busy the ride is, how many people with special needs are already waiting).