Basically, from a mechanics point of view, shields in Star Trek work like armor — it soaks some of the incoming damage.  The shields degrade as more damage comes in, eventually failing.

Shields in Cortex Trek have two ratings:  wound and stun.  The Constitution-class starship for my retcon game (much closer to the original show ship) has a rating of 3W, 3S.  An attack from a Klingon D-7 might take a shot at Enterprise. The Klingons are firing at medium range, needing an average (7) to hit.  The gunner is an NPC so the GM rolls the ship’s ALE of d8 and a Heavy Weapons of d4: they get a 9.  Big E is hit with 2 basic damage, which is split into 1W, 1S.  Both are soaked by the shields.  They roll the damage on their disruptors: and roll a 2.  Another 2W hit the shields, but do not get through.  However, the shields lose a point of effectiveness.  This starts with Stun, then moves to Wound.

Here’s where the operations players can be of assistance.  While the tactical officer is shooting, the operations (navigator usually in TOS) tests to try and keep the shields fro losing effectiveness.  They roll their Tech/Shields as if doing a wound recovery test, in this case against a 5 (and average test.)  Say the character has an INT d8 and a Tech of d6.  [S]he rolls a 10.  The shield strength fluctuates, but remains 100%.   Had he rolled under a 7, the shield rating would be reduced to 3W, 2S.

If a shot breaches the shields, the ship tests for the reliability of the shields (VIT+WIL).  Say the next shot, the Klingons get really lucky:  they roll a 12.  That’s 12-7: 5 points of basic damage (2W, 3S), and roll a 6 on the damage.  8W, 3S total.  5W gets through and Enterprise takes a hell of a shot.  The shields are checked for their reliability by the engineering officer, command officer, or if neither of these positions is a player character, they GM.  (Engineering and command officers, however, can blow plot points on the test.)  The total hit of 11 requires a 19 to succeed.  They roll the VIT+WIL of the ship — a 5 in this case.  The shields are now at 3W, 2S.

Big E can take 18 points of damage.  She has 5 hit the hull.  That damage can effect the ship’s systems.  Do a series of reliability tests for the following attributes:  Agility+Willpower for the ships’s maneuvering thrusters vs. Average (for the 5W), Alertness+Willpower for the sensors, Intelligence+Willpower for the computers, VIT+WIL for systems like transporters, warp drive, weapons.  A failure lowers the die rating a step (a botch drops it to d0.)

Example:  Enterprise takes 5W.  AGL+WIL is a 7 — no damage to maneuverability,  VIT+WIL is a 4 — there’s a problem with life support and the Vitality is reduced to d8 until it can be repaired, and so on…  For systems, a VIT+WIL for engines that fails might mean the warp drive is offline, the same for transporters.  For weapons, it might mean a die step penalty for damage as a result of power loss, damaged emitters, etc.

This is where the engineers come in.  Mr. Scott, super-engineer, has an INT of d10, Mechanics/Repair of d10, and a Talented Engineer of d6.  He has a few things to attend to — he could try to repair the shields, the life support, or another system (one per combat turn.)  This is a bypass or jury rig, since a real repair takes time.  He rolls against the damage as a wound recovery (if the damage is STUN, it’s a straight test at Average.  This assumes it’s blown breakers, and the like.)  The system gains back a die step until the action sequence is over, or the system suffers another failure — in which case the jury rig fails and the original die step penalty is restored, along with the new damage.

An engineer can stage a second wind at anytime, recovering any stun damage and restoring a system of his choice.

Alternately, the GM might want to randomly damage systems when the ship is hit. For that, here’s a quick random systems damage “chart”: