what's the system behind tactics points?

Yes, there was a change at the end of April and some tweaks along the way. 

Original change: https://www.chess.com/blog/News/chess-com-dev-update-april-28-2017

Updates: https://www.chess.com/blog/News/chess-com-dev-update-may-26-2017

There is a base rating change of +/- 10 and is modified by the time taken, number of moves, and rating difference. The algorithm also chooses from a larger range of tactics; where the old method was +/- 100 (or maybe 200) the new method produces tactics up to 500 points on either side of your rating. 

I have noticed on the more recent tactics I have done (past week or two now) that I seem to get a larger amount of lower rated tactics, over 75% being lower rated and around 50% being over 200 points lower. Most of those tactics will result in low point gain on success (as they should) and higher losses on failure (again, as they should). The main problems seems to be the mass of lower rated tactics and not a balance between higher and lower rated.

Based on your last 25 tactics, the same thing is happening to you:

I've posted a topic in Site Feedback & Suggestions about balancing the ranges.

https://www.chess.com/forum/view/suggestions/tactics-selection-algorithm

Ideally, 10% would be +200 or more, 80% would be between 200 and -200 (with about 50% of those above or below) and 10% -200 or more below. 

I don't know if the problem selection being weighted is a design decision or a bug, so for now I'm leaving it as a suggestion.

I went down 200 points since this new algorithm... I wonder if I'll end back at 0 if I keep playing like this.

I (also) wonder how much the time you take to solve a problem influences the score. Does every second count? Or is the diffrence in score only influenced by the difficulty of the problem.

It is based on the average time to solve and your rating compared to the tactic rating. So, in a problem with an average of a few seconds, seconds could influence the score.

They've thrown in an item labeled target time, which is significantly lower than the average time.  I think that this has more influence on the rating change than the average time.

I've found that even if I immediately see the puzzle's solution, I can't move the mouse fast enough to meet the target time.

Also, I've noticed that the puzzle ratings seem to be haphazard.  Today, I had five puzzles that I thought were about the same level of difficulty.  The ratings of the puzzles ranged from 1100 to 1900.  It doesn't make a lot of sense.

Target Time is the time needed to get a 100% score, that is it. If the problem is significantly lower rated than your current rating and the average time is not large, then it is expected you should be able to see the tactic solution and solve quickly. But again, that only impacts getting 100% of the possible score, which may be low due to the rating difference.

As to the tactics ratings, those are based on people solving them but the algorithm selecting them pulls from a much wider range than in the past.

Additionally, if you see you can win a Q for a R and have a winning material advantage, you should not be penalized 15 points for taking it and not seeing the tricky mate-in-three because you are being encouraged by the ridiculous target time not to look for other solutions.

 This is horrible and not acceptable at all, ios doesn't work and the Android suck if you play chess on them.

Tactics are obviously biased toward blitz players.

Tactics decide games in all time controls - from bullet, through classical and all the way to correspondence.

It's a skill well worth mastering - as it adds flair and grace to one's play, and helps a player squeeze the utmost from his pieces and positions, seeing possibilities that the ordinary player doesn't even suspect. 

There are no excuses to not mastering this essential part of the game.

Someone correct me if I am wrong but I feel like you used to lose points for taking too long but now you will gain a minimum of +1 no matter how long you take. I prefer this. It's advisable to take your time until you find the solution rather than gamble. Some you will get quickly and some you won't. 

The new system seems to be in my favour. I've jumped 200 points in the past month or so.  

I still think that there is something wrong with the algorithm that calculates the puzzle rating.  I would expect puzzles of equal difficulty to have similar ratings, not an 800 point spread.  Also, as I have noted in other threads on this topic, I see problems that have a 15% or so pass rate and have a rating of 1100 to 1300.  To me, that's extremely inconsistent.

I also have a problem with target times of 8 or 9 seconds.  That's not measuring my problem solving abilities, it's measuring my reflexes, which at 76 years old, are slow.