Lighting – What type of lighting is used in this picture

You can buy LED fixtures off the shelf, which are basically ready to go. I've seen these as puck-shaped lights that are basically designed for under-cabinet or shelf lights.

There are other pretty simple off-the-shelf solutions as well, such as using rope light.

If you are talking about using raw LEDs and literally doing everything yourself, it's pretty simple. You need to get some high output LEDs (pay attention to color temperature, beam angle, etc to make sure it is suitable). You can get LEDs in different packaging: clear/diffused, large/small, forward/side-emitting. Likely your best bet will be a large frosted one. How many you need will just depend on how much area you're covering, and how bright you want it.

You'll need a DC power supply. The exact values are all dependent on what you use, so it's very hard to give an answer here. There are online LED array calculators available, which can give you something like this:

One of the major disadvantages of serial wiring (shown here) is that if one LED is disconnected or burns out, they all go out. Finding which one is broken can be difficult, and if you've ever done this with x-mas lights you'll understand what I mean.

Another possible disadvantage of the DIY solution is that you are either going to have to come up with a nice way to package this all underneath, or you'll have a bunch of wires and bare LEDs hanging under your cabinets (maybe hidden by some moulding, but still). This may turn off future potential buyers. Off the top of my head, using some split-loom tubing may be a way to hide the wires and make it look at least half-decent.

Wow! You can use the screw in type LED bulbs, which contain their own controllers. These are typically dimmable down to about 20% using standard wiring. Here, we have Cree bulbs that look VERY similar to old Edison (tungsten) A19 bulbs, rather than the funny looking futuristic LED's. They're also much cheaper than the older screw-in LED's, about $8 each for the newest type. I don't know if they're available in your area. I LOVE THESE BULBS. ref: homedepot.com/p/Cree-40W-Equivalent-Soft-White-2700K-A19-Dimmable-LED-Light-Bulb-with-4Flow-Filament-Design-BA19-04527OMF-12DE26-3U100/205597080

If you're going with low voltage, (12V) you can use the LED strips and get rather creative, rather than using standard lighting fixtures. I replace fluorescent tube lighting with LED quite often now, for instance, or you can use them in an area with a diffuser panel below, as is typical in kitchens here. Friend of mine did the same with his bathrooms. I've also seen them used inside a paper light globe, wound in creative spirals and such, to make a unique decorative touch. Also consider using the strips as they were originally created to be used, as under-cabinet lighting. Be aware, however, you can't have a dimmer on the AC voltage side (before the transformer) and will have to use dimmers supplied for the 12V LED strips. I've seen these with remotes, and that's probably your only option for those. On a side note, the LED strips come in various color outputs (Bright white, Red/Green/Blue (for accent lighting or novelty lighting,) Warm white, etc.) The multicolor ones do require the specialty 12V controller to get all the effects.