Grafting
Grafting is a technique that vegetatively joins two plants into one - instead of cross-pollinating two plants to produce seeds, grafted plants use the roots and bottom portion (rootstock) of one plant and attach it to a tender shoot (scion) from the top portion of another plant. [1]

Grafting is one of the methods of asexual propagation, also a method to create seedless fruits. This method is a common practice in horticulture and has a long history dated back to 2000BC!
The Hebrew bible has text hints on the practice of grafting!
Romans 11:17-24 (NIV)
17 If some of the branches have been broken off, and you, though a wild olive shoot, have been grafted in among the others and now share in the nourishing sap from the olive root, 18 do not consider yourself to be superior to those other branches. If you do, consider this: You do not support the root, but the root supports you. 19 You will say then, “Branches were broken off so that I could be grafted in.” 20 Granted. But they were broken off because of unbelief, and you stand by faith. Do not be arrogant, but tremble. 21 For if God did not spare the natural branches, he will not spare you either. 22 Consider therefore the kindness and sternness of God: sternness to those who fell, but kindness to you, provided that you continue in his kindness. Otherwise, you also will be cut off. 23 And if they do not persist in unbelief, they will be grafted in, for God is able to graft them in again. 24 After all, if you were cut out of an olive tree that is wild by nature, and contrary to nature were grafted into a cultivated olive tree, how much more readily will these, the natural branches, be grafted into their own olive tree!
Why Grafting?
There are a variety of purposes to use the grafting technique [2][3]:
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To retain or change varietal characteristics
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To repair injured trees
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To strengthen plants’ resistance to certain diseases
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To ensure and optimize cross-pollination and pollination
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To produce multi-fruited or multi-flowered plants
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To increase the growth rate of seedlings
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To produce perpetuate clones
Read more:
[2] Grafting and Budding Nursery Crop Plants (Link)
[3] Graft by The Editors of Encyclopaedia Britannica (Link)
This Crazy Tree Grows 40 Kinds of Fruit | National Geographic
In the video below, Sam Van Aken uses "chip grafting" to create trees that each bear 40 different varieties of stone fruits, or fruits with pits!!
Funny Graft
As I was browsing about grafting, I saw gardeners graft tomato with potatoes - tomato growing on top and potatoes at the bottom! They name it as a 'ketchup-and-fries' plant! (Photo Credit: SuperNaturals Grafted Vegetables, LLC)
Grafting Methods
There are different names of methods in grafting over different YouTube videoes. It may be confusing if you only begin to learn about this technique.
Let's look at the graphic below from Scientia Horticulturae 127(2):93-105 which summarizes major grafting methods in cucurbits (e.g. squash, pumpkin, zucchini) and solanaceous (e.g.potatoes, tomatoes, eggplant, capsicum, chillies) vegetables [4].

Major grafting methods in cucurbits and solanaceous vegetables:
(A, B) hole insertion grafting;
(C) tongue approach grafting;
(D, E, J) splice grafting;
(F, G) cleft grafting;
(H, I) pin grafting.
These two videos demonstrate how actual grafting happens.
To learn more, here are a very good Grafting lessons series by Skillcut - total 10 sessions !
Reference
[1] What Does Grafting Mean? (Link)
[2] Grafting and Budding Nursery Crop Plants (Link)
[3] Graft by The Editors of Encyclopaedia Britannica (Link)
[4] Lee, Jung-Myung & Kubota, C. & Tsao, S. & Bie, Zhi-Long & Echevarría, Paulo & Morra, Luigi & Oda, M.. (2010). Current status of vegetable grafting: Diffusion, grafting techniques, automation. Scientia Horticulturae - SCI HORT-AMSTERDAM. 127. 93-105. 10.1016/j.scienta.2010.08.003. (Link)

