Highlighting certain area with color, leaving the rest grayscaled — what is the name of this effect in photography?
What is the most precise way to achieve this effect in Lightroom?
I'm surprised at all the incorrect answers to this! The technique of converting part of a photo to B&W is known as selective colour (the resulting image is sometimes referred to as a cut-out). There are two ways of doing it in Lightroom (v2 onwards) depending on the effect you're trying to achieve. They're both very easy to use. I'll demonstrate using this as an example image:
Method 1: Leave a specific area in colour
This is done using the Adjustment Brush.
Convert to B&W but leave certain colour(s) in colour
This is done using the Hue/Saturation/Luminance (HSL) controls.
I agree that Photoshop might produce better results: simply desaturating isn't always the best way to convert to B&W, and Photoshop gives you much finer-grained control over the process. But you asked about Lightroom and it's definitely possible. Both of the above examples took a few seconds each: with a bit more care and attention they could both be improved.
'Selective color' can be somewhat achieved by using the adjustment brush and brushing saturation '-100%' out of all the areas you don't want to have color. This would be CRAZY HARD for many things because you can't mask areas off in Lightroom. Really, Lightroom is the wrong tool for it.
Depending on the image, you may also be able to drag the saturation for individual color sliders down to achieve a similar effect, but the results are unlikely to be great.
For this, your best bet is PS or the Gimp.
(not having LR experience, and not being a PS expert) in PS you would use a layer'ed approach. In your first example creating a BW copy of your image and using a layer mask to expose where the color version popped through. A similar layer/mask approach altering the intensity of the image in your second example.
I am not sure if LR supports this sort of layer editing, but googling for LR and layers shows up lots of results for a plugin that seems to be able to do this.
First link I found on google (and I repeat I am not endorsing this product) adding layers to lightroom