Engineers were the last of the ship types (in version 2.8 anyway) to be implemented. They are one of the most powerful technologies available, but they can get you killed as fast as they can help you win. Their high expense means that you need to know what you're doing with them so as to not weaken your forces or use too much tech development. On the other hand, there are those like Brad Chase, a.k.a. Ah, who have made a (disgustingly regular <grin>) habit of using 'neers to demolish veteran players.
How Engineers Work
The maximum BR of an engineer decreases each time it is used. The amount of this decrease is set by the server administrator, but is generally either 1.0 or 0.5 levels of BR per use. Note also that an engineer must have a BR of equal to or greater then that value in order to open or close a gate. Therefore, if you plan to open/close a gate in conjunction with either a battle or an overbuild, then you should make sure to to use a higer BR-level ship.
The cost of building and maintaining engineers makes high-BR ships prohibitively expensive. Most players produce BR1-3 engineers and then just make a new one if another is needed a few rounds later. This, though, means that you have to be sure to protect your engineers — keep a fleet between them and the enemy fleet, or even better use them where they cannot be watched.
The main purpose of engineers is to open and close jump gates between systems. This has both offensive and defensive capabilities — you can open a gate to attack through, or you can close a gate to keep an enemy out. You can even open a jump to a planet that you have not explored yet (just have a science ship of your own handy to send through the new jump as soon as it is opened.)
The basic defensive use of the engineer is to keep enemy ships from going someplace you don't want them to go. This may range from a desperate effort to keep your home world from getting nuked, to shutting off access to rear planets. This last use is particularly important when your opponent is using cloakers — if he can't get them there you don't have to worry about watching for them or defending against them. (On the other hand, do keep watch over closed off areas to make sure the enemy hasn't opened his own jump into them.)
Note that the opening and closing of jump points should not be limited to simply allow your fleets to travel. Creative opening and closing of specific gates will allow you to organize the map as you see fit — i.e. so as to be most strategically advantageous to you. Not only can this prevent the enemy from exploring certain of your planets, but it forces him to attack through a specific corridor that you have established, and that favors you in geography.
The 'Neer Trick
The most obvious offensive use for engineers is simply to open a shortcut alling your ships to reach the enemy faster. Given the right map setup, though, this turns into the classic 'neer trick. This is where you can open a link into your opponent's backfield, but your opponent cannot see your engineer or the fleet with it. You can then pour your fleet through the gate, taking a turn to explore with a science ship if you can't already see the other side, and usually nuke the planet (optimally, his Home World.)
A derivative of the 'neer trick can be used when the opponent *can* see your engineer and fleet. In this case you have stationed a small fleet of cloakers on his side of the jump, and you uncloak them as your fleet comes through. Your opponent will probably not have built enough to account for your cloakers, and so most of your fleet will survive to attempt a nuke the next turn. Alternately, you might show him only a small fleet with the engineer to make him think you are going somewhere else with them, while the bulk of your strength is in hidden cloakers sitting over his planet.
Beware of the 'neer trick, though, lest it backfire on you. An experienced player can usually tell where your builders are by how long it takes your ships to reach the front. If he can anticipate your opening the gate he can be waiting for you to open the jump, and will have a science ship set to explore your planet. Remember that opening a jump exposes your planet as well as offering you access to your enemy's planet. It is often wise, then, to open a jump from a small planet rather than a builder or Home World.
Another danger of opening a jump is that doing so gives away information about your map. While it isn't necessary, I like to always know what is behind the jump I'm opening. If your attack is repelled, your enemies knows where they can open a jump for future attack, even if you close the jump right away. If he manages to explore even that one planet on your side of the jump before the science ship is destroyed, he can therefore also see he links from that planet and thereby know if there are other nearby jumps that he can open without you seeing it.