How to Trace a Raster Image into a Vector (SVG) Using Inkscape
First, install Inkscape from inkscape.org.
When you open it up, it should look something like this:
Open the "File" header, select "Import...", and select the image you want to trace:
A new window will pop up, asking about import settings. You should be able to leave them as is.
This is what the document should look like once you've imported your image.
Select your imported image.
Open the "Path" header and select "Trace Bitmap...".
At this point, a window will pop up to allow you to configure the trace. Make sure to turn on the Live Preview. If you are tracing a single-colour (and shade) image, "Brightness cutoff" will work. Otherwise, use "Colors" and adjust the Shades value.
Once you're happy with the trace, click "Ok" once and close the window. You should now be able to drag the trace off the base image.
You can tell which one is the trace by mousing over it with the Node tool (second from the top on the left):
Select the base raster image again:
Then delete it:
Now is the time to clean up any speckles or extra bits that may have been caught when tracing. Select the Node tool again, select the trace, and select any nodes on the trace you want to get rid of.
Select node(s) you want to get rid of, and delete them. (in this case there weren't any speckles, but it's just for demonstration)
Once you're done, click off the image and click on a solid part to select the whole trace.
Open the "Edit" header and select "Resize Page to Selection":
By now, the trace should be done. Make sure the trace is still selected.
Finally, open the "File" header, select "Save As...", and save it.