How to Trace a Raster Image into a Vector (SVG) Using Inkscape

First, install Inkscape from inkscape.org.

The demo image is here, and sourced from freepik.com.

When you open it up, it should look something like this:
A new, blank Inkscape document.

Open the "File" header, select "Import...", and select the image you want to trace:
A blank Inkscape document with the "File" menu open and the "Import..." button selected.

A new window will pop up, asking about import settings. You should be able to leave them as is.
An Inkscape document with the import settings open.

This is what the document should look like once you've imported your image.
An Inkscape document demonstrating what it looks like after importing an image.

Select your imported image.
An Inkscape document with the newly-imported image selected.

Open the "Path" header and select "Trace Bitmap...".
An Inkscape document with the "Path" menu open and the "Trace Bitmap..." button selected.

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.
A screenshot of Inkscape bitmap-tracing settings.

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.
A screencap of Inkscape after the trace has been dragged 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):
An Inkscape document with the trace highlighted in red.

Select the base raster image again:
The document with the base image selected.

Then delete it:
The document with the base image deleted.

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.
The Inkscape document with the trace selected with the Node tool (showing each node in the path).

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)
The document with a node covered by a selection box. The document with a node selected. The document with the previous node deleted.

Once you're done, click off the image and click on a solid part to select the whole trace.
An Inkscape document with the trace selected.

Open the "Edit" header and select "Resize Page to Selection":
An Inkscape document with the "Edit" menu open and the "Resize Page to Selection"

By now, the trace should be done. Make sure the trace is still selected.
An Inkscape document with the canvas cropped down to the trace size.

Finally, open the "File" header, select "Save As...", and save it.
An Inkscape document with the "File" menu open with the "Save As..." button selected.