You also need to know how long it needs to run in order to calculate how many batteries you need.
The bare LED (often called the emitter) has a metal plate on the back side. Some have only one for heat dissipation, some (including both that you mentioned) have two additional pads on the back for electrical connection. On some emitters the electrical connections are on the front. When they are all on the back, the trick is to get the center pad really well connected to the heatsink, and still get the electrical connections out. I've seen pictures of special copper mounting brackets that have a narrow portion in the center where only the center pad is soldered directly to the copper, and the electrical connections hang off each side, so wires can be soldered to them. Its really an awesome way to do it.
But most of us haven't the will or ability to do that, which is where the star comes in. It's a special PCB with metal core (aka MCPCB). All three pads of the emitter are soldered onto it. The thermal pad had very good (but not awesome) contact to the metal core, which transmits the heat to your heatsink. The electrical pads are brought out to the sides so you can solder wires to them. If you aren't going to design your own MCPCB or fancy copper heatsink, the star is about the only way to go, other than buying a module with emitters installed.
A driver IS a constant current power supply. I make and sell a very simple one based on an old and well-known linear constant current circuit. Unfortunately it doesn't play well with LiIon cells. Ones that do are usually switching supplies. They are more sophisticated and if well designed can keep the emitter safe and running at or near full brightness for most of the battery life. But as with anything else, there are good ones and bad ones, and some in between. Good ones should be reliable, efficient, and deliver the promised performance. Bad ones tend to be less efficient, less reliable, and offer performance that just doesn't measure up to claims. Often the output isn't as 'constsnt' as you'd like as the battery voltage or LED voltage changes, even within the specified range. Good ones will be rock solid, or pretty close.
Efficiency is a measure of how much of the battery power the driver sends to the emitter. The driver will always consume some of the power as it passes through, converting it to heat. A good driver can have an efficiency of 90% or even higher, while cheap ones tend to be in the 60-80% range. A good driver can increase your battery life 30-50% over a bad one, at the same brightness.
The best drivers are in the $30-40 range, the cheapest are under $5. But cheap and bad are not always the same. There are some very inexpensive drivers that don't suck. But if you're paying $1.99 for the driver and $2.89 for shipping, you are taking a chance on what you'll get. At least it's a cheap gamble.
