Resize a selection to specific pixel dimensions?

If the object/selection is on it's own layer, make it a Smart Object

Double click the SO to edit in a new window and use Image Size to resize it, and Ctrl S to save back to main document window

When I right-click on the selected region and select Transform, the tool settings at the top shows 600 x 1426  -- what do these numbers mean?!

By @peternelson

The numbers mean what they say, and although that sounds like a bad answer, the point is to look more closely at what it says:

You said it shows 600 x 1426, but that's not all it shows. What it actually shows, if you look closer, is:

X: 600 Y: 1426

So, those are X and Y position coordinates.

X = horizontal position

Y = vertical position

If you change those numbers, you’ll reposition the selected pixels.

You want to resize. That is the next set of numbers, which in your screen shot show:
W: 100%

H: 100%

W = width

H = height

Of course, you don’t want percent, you want pixels. So follow the advice in Kevin’s reply, and type the number followed by “px” to tell Photoshop you want to set the size in pixels. If you want a field’s value to be in pixels all the time, right-click the W or H value and choose Pixels, or to set the unit of measure for almost all fields at once,  right-click the ruler and choose Pixels.

If you put together the advice in all of the replies so far, these are some general tips you can take away:

• Get in the habit of checking the options bar and Properties panel, because that’s where most settings can be changed for a tool or selection. 
• Look at all the numbers, and pay attention to the labels next to the numbers. 
• You can often get to more/hidden options for something by right-clicking it. 
• If you don’t know what you’re looking at and the labels don’t help, hover the pointer over the option to see the tool tip for it. 

@peternelson when transforming a selection the 600 x x 1426 y are the coordinates for your current selection size. If you look a little more to the left you will see the matrix showing what point of origin the value is coming from. Be default its the center of the selection:

To the right of that is the w & h values that you can enter to resize your image. It will show % but you can enter px values:

Here is how I would do this. Start with the selection tool active and no selection.

In the options bar, you can change the Style to "Fixed Size" and then enter your values.

Now when you use the tool it will auto select a size of your choosing.

If it's selected, then you can enter the value you want to change its dimensions in the Transform section of the Properties panel (Window > Properties).