Welcome to the Forum. I'm really looking forward to reading you posts and threads on various topics on Martial Arts etc
Sandan is third Level, Nidan is Second Level
In Knockdown Karate Circles A karateka is known as Sempai (Senior) upto Sandan whereas from there the Karateka is known as Sensei.
Regards to Karate and what it means personally
From the Mudansha/Yudansha grades right through to Nidan one is going through the motions learning kata etc concetrating on The Grading Syllabus etc. But when one reaches Sandan it is like a breath of fresh air, one no longer needs to worry about gradings (many styles only grade to Sandan), one can concetrate on the Ryu, ones Karate itself, there are no longer any timescales, no pressure on getting to a certain level by such a date etc. one can concetrate on a particular technique for as long as it takes to feel this as a natural being.
By Sandan one must be nearing the Ri part of "ShuHaRi" of you and your Karate becoming one. A Kata is not just a "dance" but a way of life. One appreciates ALL Budo and humility of others higher and lower than oneself are instilled into this Level.
I always say that a Mudansha needs to be advised what to do, but a Yudansha already knows what to do.
This can be used for all Dan Grades but I'll expand as to what I need to see before I put Student forward for Shodan (1st Dan)and above is I'm looking for mental attitude as the physical is only part of it.
Seven virtues of Bushido
The Bushido code is typified by seven virtues:
Care for the aged
You can teach a monkey to perform karate (you can find this on You tube), does that mean it is worthy of a black belt. To me and many Yudansha (blackbelts) the above Virtues are what makes a Yudansha not the physical side.
A man is but the product of his thoughts what he thinks, he becomes.