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How to draw a heatmap with grids of different colors in a map?

Recently I have been reading a paper called Modeling Taxi Drivers’ Behaviour for the Next Destination Prediction. There is a figure(Fig.1) that I wonder how to draw. Based on what I know, it may be drawn by Python. Then what library of Python should I use to draw such a heatmap?
Thanks a lot in advance for your time and your expertise.
Best Regards

I have built something very similar for myself in the past. However, as you are just wondering how this would be drawn and not exactly asking how to do it with Python, I will share how I did it.
1. Building Grid:
The grids on the map are squares of X-size latitude and longitude. In short, those are latitude and longitude grids. I used an interactive map library named leaflet.js to build the world map with an overlay of latitude and longitude grids. This is the tutorial I followed from the leaflet: https://leafletjs.com/examples/choropleth/
Remember, you have to and can build your version of the grid to overlay on a world map using GeoJSON as discussed in the tutorial. At least, when I was building, there wasn't a publicly available version of the lat/long square grid.
2. Showing Colors (Heatmap):
Once you build the grid with GeoJSON, the leaflet can take the whole GeoJSON as it is and overlay it on any map of your choice. That means you can put the numbers(aka data) for each grid in the GeoJSON. This part is also shown in the same tutorial.
For my project, I used to create a complete GeoJSON formatted file with normalized data in Python and then visualize it in leaflet.js. Below is an example of what I have built using these tools.

How do I plot a contour plot of 3 column data

I have a 3 column irregular data in the format [X Y Z]. I am having difficulty in creating a contourf
plot for the same since it requires one to create a meshgrid and also that data be uniform among the grid.
I need some direction or hint to get started.

I am providing two ways in which you can create a contour/density plot for the data which is in 3-column format and irregular, as you have mentioned.
You can use Mathematica: see the documentation of ListDensityPlot. You can directly provide the data as, ListDensityPlot[{{x1,y1,f1},…,{xk,yk,fk}}], and this will plot the sought density plot.
There is also a simple way to do this in python: You can see the documentation of tricontourf, a module of matplotlib. Its functionality is similar to that of contourf, except that you give 1D arrays rather than the data in mesh grid format.

How to slice 2D grid from 3D irrigular data?

I have some data of soil's moisture content (Theta) in the form of 3D-domain points (CSV file of the columns x, y, z, Theta). I want to take cross sections from the 3D domain in some specific positions (section ABCD in the figure). I want to calculate the value of Theta in a 5*5 grid in the cross-section, but the points around each node of the grid are not coplanar with the unknown point. I did this before for 2D domains in python, but the 3D domains seem more complicated for me. I found that plotly can make something like that in its virtual environment but I want this to output a numpy array or pandas DataFrame to draw it as a contour in the jupyter notebook.
I know that finding the grid involves finding the value of each point like P0 in the figure by interpolation or gridding from its neighbors, then to draw the cross section using matplotlib, but I don' know how to do it.
Related question, Is slicing 2D grids from 3D grids available in matplotlib or similar libraries?
Thanks for all help.

The underlying problem is 3D interpolation. There are numerous packages which can do this type of thing, or you can write your own (using, e.g. KDE, which is basically just a type of smoothing/binning). There is a lot of material on the topic, like
This answer https://stackoverflow.com/a/15753011/230468
The scipy docs
This extensive set of option on scicomp.stack
And this blog post (with some good examples)

Have you tried playing with pyugrid? It's a library specifically for manipulating unstructured grids, so it sounds like it might be of some use to you. Check out these example notebooks.

Best way to create a 2D Contour Map with Python

I am trying to create a 2D Contour Map in Python that looks like this:
In this case, it is a map of chemical concentration for a number of points on the map. But for the sake of simplicity, we could just say it's elevation.
I am given the map, in this case 562 by 404px. I am given a number of X & Y coordinates with the given value at that point. I am not given enough points to smoothly connect the line, and sometimes very few data points to draw from. It's my understanding that Spline plots should be used to smoothly connect the points.
I see that there are a number of libraries out there for Python which assist in creation of the contour maps similar to this.
Matplotlib's Pyplot Contour looks promising.
Numpy also looks to have some potential
But to me, I don't see a clear winner. I'm not really sure where to start, being new to this programming graphical data such as this.
So my question really is, what's the best library to use? Simpler would be preferred. Any insight you could provide that would help get me started the proper way would be fantastic.
Thank you.

In the numpy example that you show, the author is actually using Matplotlib. While there are several plotting libraries, Matplotlib is the most popular for simple 2D plots like this. I'd probably use that unless there is a compelling reason not to.
A general strategy would be to try to find something that looks like what you want in the Matplotlib example gallery and then modify the source code. Another good source of high quality Matplotlib examples that I like is:
http://astroml.github.com/book_figures/

Numpy is actually a N-dimensional array object, not a plotting package.
You don't need every pixel with data. Simply mask your data array. Matplotlib will automatically plot the area that it can and leave other area blank.

I was having this same question. I found that matplotlib has interpolation which can be used to smoothly connect discrete X-Y points.
See the following docs for what helped me through:
Matplotlib's matplotlib.tri.LinearTriInterpolator docs.
Matplotlib's Contour Plot of Irregularly Spaced Data example
How I used the above resources loading x, y, z points in from a CSV to make a topomap end-to-end

Plotting graphs with error ribbons in python

for a while I've been trying to come up with a good way to graphically represent a data series along with its estimated error.
Recently I saw some graphs where the data was plotted as a line, with a background 'ribbon' filling the area between the lines plotting data +/- sigma.
Is there a name for this type of graph, and is there any python toolkit which has the capability to make such plots?
A simple way to fake it with matplotlib would also be useful - right now I'm just plotting three lines, but I don't know how to fill the area between them.

I would use the fill_between method. Look at the Our Favorite Recipes section of the manual for matplotlib for some good examples. They have one that looks like this:
and another that looks like this: