Christian belief is that the presence of God brings life and joy. The closer you are to God, the more joyful you are, and the more "life" you have (life here not in the sense of mere existence, but in partaking of all the good things life has to offer). God is more present to those who are close followers of him, who therefore have much joy (as Paul describes) and abundant life (as Jesus says). God is present to a lesser extent with people who are not followers but have not rejected him, or who still have a chance to become followers (i.e. everyone who has not yet died).
On death, a person will either be accepted by God and become closer to him, bringing joy and life; or they will be rejected by God, resulting in a separation from God, bringing despair and not-life (not-life being in the sense of still continuing to exist but having no enjoyment of the things that life has to offer). This is what separation from God means,and is also what the Spiritual Death means. A person after their physical death either has Spiritual Life with God, or Spiritual Death without.
The above is not universally believed in Christendom, but I wrote from the point of view of the people who use the term "Spiritual Death" frequently. And many other Christians believe something very similar.