Vector and raster are the two basic models for any digital representation of spatialized data. Let's see what they correspond to!

What is a raster?

In its simplest form, a raster is an image. It's even exactly what we imagine when thinking about a digital image: pixels.
A raster is a grid, also called a matrix, composed of cells organized in rows and columns (like a table). Each cell in this grid is a single pixel. Each pixel has values (values to define its color for example). It is the combination of all pixels, each at a certain place in the grid space, that creates an image.
A raster is a matrix of pixels

Since a raster is an image made up of pixels, if you enlarge the raster (if you zoom in on the image), the sharpness of the image deteriorates. At high magnification, we see the pixels. As a result, visually, the details that could be seen on the image at its original size become more difficult to distinguish.

Zoom on the blossom in raster mode: we see the pixels

The resolution depends on the number of pixels
The resolution of a raster (of an image, therefore) depends on the number of pixels it contains. The more pixels in an image, the better the resolution and visual quality. This is why the higher the resolution of the raster, the raster the file is heavy. And this because of the larger number of pixels, because each additional pixel is an additional element to store for your computer.

Raster in geographic information
Format. There are specific extensions that can be used for raster with a spatial component (geolocated raster), for example, GeoTIFF, which is a standard adding georeferencing information to a TIFF image.
Data type. Raster are used to represent continuous data and discrete data (= discrete, distinct).
In raster format, the continuous data can be altitude data for example (each pixel, in this case, has a value of elevation/altitude). Continuous data can also be spectral data (wave spectrum, not Halloween spectra). 
Make Raster to Vector your Image
What is a vector?

The vectors are not composed of pixels. Vectors are "mathematical drawings". We also speak of vector drawings or vector images.
A vector has two components: an attribute component (the information associated with it) and a graphics component (its shape, its geometry).
The attribute component of a vector is descriptive. A vector is linked to the information of its own. For example, for a vector representing a forest, one could find information such as the name of the forest owner, the age of the forest, the date of planting or the type of majority tree species. 
In its graphical component, a vector is composed of nodes (points in space) and mathematical formulas for calculating the arcs (lines) which connect these nodes to each other and which thus form a geometry.
There are three main types of geometry: point, line, and polygon. If a vector is composed of a single node, a single point: the vector is a point. If a vector is composed of several points connected to each other, but not forming a closed geometric shape: the vector is a line. Finally, if a vector is composed of several points connected to each other, and forming a closed geometric shape: the vector is a polygon.

The three basic geometries: point, line, and polygon

Infinite size for a featherweight
The minute math: do you remember how to calculate the area of a square? It is next to the square x side of the square, is C x C = C 2 . To have a larger square, we increase the value of the C side. There is no need to store thousands of additional pixels, only the numerical value of C.
Consequence: the size of a vector file does not very much according to its size, the computer only needs to keep in memory the value of the numbers composing the equation which defines the vector image (which takes up relatively little space) and not tons and tons of pixels.
A vector is a calculated object, so it is possible to zoom in and out of the image, without altering its visual quality. Since a vector image is composed of equations that represent points connected by paths, its size may well become gigantic: you will never see a single pixel, the edges of a vector remain smooth.
Zoom on the blossom in vector mode: the drawing remains smooth

Vectors in geographic information
Format. A well-known file format for geographic information is the "shapefile", better known as the shapefile. This format makes it possible to store geometries (points, lines, polygons ...), their geo-locations (their coordinates in the space) as well as the textual attributes related to each vector.
Data type. The vectors make it possible to represent discrete data (discontinuous, distinct), also called thematic data. This can, for example, be a shapefile with land use data ... in vector format. Each vector is then a distinct object (a body of water, a beach ...) having its own geometry, its own geo-location and its own attributes (its identifier, its name, etc. ).

In short, what is the difference between raster and vector?
A raster is an image composed of a matrix of pixels, each pixel having its own values. A vector image has a graphical component composed of points connected by arcs, all calculated by equations, and an attribute component composed by the information associated with the vector.
A raster represents an image; a vector represents an object. A vector is calculated, so it requires some processing from your computer to be displayed, but little memory to be stored. A raster is mostly a big pile of pixels, it requires less processing but more memory.
A blossom in vector mode on the left and in raster mode on the right

The raster, these unloved ones ...
Finally, note that the manipulation and analysis of geographic data are very different depending on their structure (raster or vector). The analysis of vector data is less complex than the analysis of raster data; the latter requires specialized skills and tools, and in particular uses the sciences and technologies of physics. And this is certainly what is good for raster to very often serve as background, support for vector data that are superimposed over ...
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