I have barely ever talked about this to anybody and most people didn't fully understand when I did, though they were gracious about it, but I will share with you my experiences with what I for myself have always called crying in the Spirit.
The first time this happened to me was shortly before (yes,before) my conversion. I was reading a small booklet, entitled Gott finden, aber wie (How to find God) and it caused me great inner conflict. When I got to a certain passage in the book, I suddenly broke down crying harder than I have ever cried in my entire life. It was a loud crying (though I tried to suppress it in order not to alarm anyone in the house), kind of like when a child cries, and it shook my whole body. The stunning thing about it was that I seemed intellectually, and even emotionally, detached from what was happening. It was as if I was watching myself, wondering what on earth was happening. There was nothing in that passage in the book that touched me on any conscious level, but the crying seemed to come up from deep inside of me. At that point I knew nothing of a person's spirit and little about the Holy Spirit.
It lasted for maybe a minute or two and stopped as abruptly as it started. When I read the book a second time, the very same thing happened to me again at the same passage. This was not something that I had any control over and it convicted me of the fact that what I was reading was most likely true. It contributed to my repenting before God and placing my trust in Jesus only days later.
Since then I have experienced this type of crying several times, always in prayer. I'm now less detached from it and it enables me to literally cry out to God for the things he lays on my heart at this point, yet there remains a feeling of being well and able to think independently, despite what would from the outside look like I am having a complete breakdown. I believe if anyone could watch me they would be truly concerned for me, and stunned when they see me return to normal and smiling with basically one deep breath. I can not initiate these episodes and I'm not in control of it, yet I actually welcome them. I know that they come from or through the Holy Spirit, since they draw me into God's presence and always lead me to more prayer, to repentance and a renewed commitment to serve Him.
On a side note, back when I decided to believe the Bible and prayed to receive Jesus as a result of reading that book, immediately after finishing my prayer, I laughed in the same manner that I had cried before, really hard, despite that there was obviously nothing funny, completely detached and thinking, O boy, that's it, I've lost my mind. That has never again happened afterwards. Maybe some day? God is so good, praise His holy name!