Just in case there are others looking to solve this, I'll put the solution up!
Different Solutions, for different cases
1st
- Click once on the picture to select it
- On the toolbars or ribbon look for the 'Borders' control
- If it's highlighted, click it to remove the border
If that didn't solve it for you
2nd
- Right-click the picture
- cChoose 'Format Picture'
- Go to the 'Colors and Lines' tab
- In the Line section
- Change the 'Color:' choice to 'No Colour'
And if THAT doesnt solve it for you, then the image must have had a border BEFORE being pasted in word, so
3rd
- Click once on the picture to select it
- You should now see (2003) a floating toolbar called 'Picture' (2007) have some new tabs in the ribbon to play with
- Look for the 'Crop' symbol (it's like two corners, overlapping - if you hover the mouse over the icons, you'll find it in the end)
- Choose this tool
- Now, when you grab the picture handles you will find that, instead of resizing the image, it cuts bits off
- Remove the frame!
- (The only problem here is, Word isn't the most accurate cropping tool in the business - so, if you need a bit more finesse, I suggest you move it into a graphics editor at this stage!)