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:

Example image:

Method 1: Leave a specific area in colour

This is done using the Adjustment Brush.

1. Switch to the Develop module.
2. Select the Adjustment Brush (right-hand panel, in the little tool strip right beneath the histogram).
3. Ensure all the sliders are at their default (middle) position, then slide Saturation right down to -100.
4. Choose an appropriate brush size, feather and flow and disable the Auto Mask feature.
5. Simply paint in the grey areas.

Result:

Convert to B&W but leave certain colour(s) in colour

This is done using the Hue/Saturation/Luminance (HSL) controls.

1. Again in develop module, open the HSL / Color / B&W pane in the right-hand panel.
2. In the top strip, click Saturation.
3. Using either the sliders or the drag control (the little circle in the top left of that pane), desaturate the picture colour by colour. I've desaturated everything except reds and yellows.

Result:

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.