A mini LP is a CD version of something that was originally released as a 12" (12 inch) vinyl LP.

In many cases the packaging is superior to, or at least more elaborate than the majority of vinyl releases. The reason being that many follow-up vinyl releases were inferior to the first vinyl release (or at least lacked some of the extras found in the first release - such as postcards, posters, lyric sheets etc).

They are often referred to as "cardboard sleeves" and "paper sleeves" being made exclusively from paper or cardboard (other than the CD of course).  Other names and abbreviations include: "jpn lp sleeve cd" and "kami-jacket". (The last of these I guess is an indication that they are held in high esteem by many who collect them.)

They are also known as "LP replicas" as they tend to exactly replicate the original packaging of the first vinyl release.

LPs (short for Long Play) are still made today.  They are made from vinyl (acetate, whatever) and are placed flat on a device known as a turntable.  This rotates the disc at about 33 and 1/3 revolutions per minute.  A needle (or stylus) is placed on the "vinyl" and the vibrations detected by this are translated into an electronic "pulse" which in turn is converted and amplified into sound waves.