The Holy Spirit is sometimes described as Jesus' Ghost, only it's not, but if you ask someone to explain it to you they'll have a hard time because it represents a power that has no corporeal form but still exerts a sense of religious awe and wonder in people when they are overcome or visited by it.
During biblical times (during the first thousand years of the church, specifically), a huge amount of reasoned consideration was applied to understand how the Holy Spirit, the Father, and Son were all one God but different persons. Ultimately, the most rational clergy chocked it up as a Mystery (gutless wimps!) and moved on to more easily explained physical phenomena, leading to the birth of science in Europe. This was ironically a good thing, because if Christian philosophy and science had never diverged, truthiness would be recognized as bad by church-goers.