
Spot the Difference
First; let's look at the differences between Vector and Raster:
> Vector elements consist of mathematical X and Y coordinates called Anchor Points, and the space between those interconnecting points is called a Path.
These Points and Paths are easily produced using the Pen (for precision Points) and Pencil (for flowing Paths) tools in Adobe Illustrator.
> Raster elements consist of pixels which are flattened across the bounding box of the entire file constraints.
The quality of a Raster element is typically determined by its DPI value (dots per inch).
This means an image with a value of 10 DPI will stretch its ten pixels across the same square inch of space that an image of 300 DPI would stretch its three hundred pixels across.
The latter of which would render a sharper and denser print.
Delving Deeper
With the aforementioned in mind, it’s worth noting that Vector elements can span infinitely to any size; as the Points and Paths that make up the entire artwork is mathematically calculated (automatically, by your software) as the artwork dimensions are adjusted. Whereas a Raster element which is increased in size will not scale its DPI accordingly; and as such will lose quality (DPI) in proportion to its scale.
> The space enclosed within a closed set of Points and Paths is called an Object.
An Object can be manipulated, contorted and converted to any colour with tremendous ease within Adobe Illustrator.
> Each individual pixel within a Raster element has a set of colour values defined by the Colourspace you are working in. RGB will consist of Red, Green and Blue colour data; whereas an image within the CMYK colourspace will be composed of Cyan, Magenta, Yellow and Black channel data.
A Gentle Reminder
We recommend a minimum of 300 DPI for Raster images.
Click Here for our guide on how to discover the DPI value of an image.

File Formats
The main file formats used for Raster elements are:
TIFF/TIF - Tag Image File Format
JPG - Joint Photographic Experts Group
PSD - Photoshop Document
GIF - Graphics Interchange Format
PNG - Portable Network Graphic
None of the above formats can contain Vector elements.
The main file formats for Vector elements are:
EPS - Encapsulated Postscript.
AI - Adobe Illustrator (document)
CDR - CorelDraw (document)
SVG - Scalable Vector Graphic
PDF - Portable Document Format
All of the above Vector formats can contain Raster elements alongside Vector elements; this means a file saved in PDF format does not automatically convert Raster to Vector and the contained element will be saved in their original format; unless otherwise converted (see next thread when you reach the bottom of this thread).

Fig. 1 - Our Test Subject
Our Live Subject
Let's take the graphic Fig. 1 as our example:
Download Here
This is an example of both vector and a rasterised image contained within a PDF document.
Within this file we can clearly see the quality difference between these Raster and Vector elements.
Ideally; we want to be working in Vector where possible.
Yet; depending on the properties of any particular artwork, it may not always be feasible or even possible to convert Raster to Vector due to the vast range of individual colours present within even the simplest image; See Here for our guide on how to discover the colour count of an image.

The Keen Eye
By taking a closer look at the image, we can easily discern the Raster elements from the Vector elements due to the visibly sharper, denser and much cleaner nature of a Vector element.
The usage of low quality Raster elements will result in low quality prints.
Common Pitfalls
Saving a low DPI Raster image within a Vector supporting file format will not convert the Raster contents to Vector.
The result is simply an embedded Raster image; and the resulting print is near identical regardless of format.
Exporting an image as PDF from Microsoft Word embeds the element at it's scaled size.
The result is a significantly diminished resolution, which is especially alarming when attempting to extract the image at a later date.
Microsoft Word or Microsoft Paint are largely unsuitable for any printing requirements.
The Software
We recommend the following programs for the best results:
Adobe Illustrator for the construction, conversion and editing of Vector Artworks; and for the arrangement of pages.
Adobe Photoshop for it's wide range photo editing tools; and specialist actions/automated workflow capabilities.
Affinity Designer is a one-off purchase which isn't outrageously expensive; and contains a huge range of Vector editing and construction tools.
Honorable Mentions
GNU Image Manipulation Program is a freely distributed photo editing software; and is a great starting point in addition to offering unique tools such as seamless tile mapping.
Inkscape is a freely distributed vector editing software; and whilst we have no experience of using it ourselves, we do hear a lot of great reports on it's capabilities.