This is something I get asked a lot and there are so many answers, the cloth, the person making it, the way it's made, the way it's cut, the way it's fitted.... And more.
To me, one of the most important things is the canvas. A bespoke suit has what's called a "Floating Canvas". The canvas is what gives a suit the shape and form it requires at the front. A RTW or MTM suit will have a canvas glued onto the forepart, this is fine for the first few weeks, but the moment the suit is dry-cleaned, the chemicals will often cause the canvas to come unstuck leaving an unsightly finish.
In Bespoke suits the canvasses are padded, stretched, shrunk, shaped and pressed until they are the exact shape the tailor requires. It is a series of processes that will an apprentice may spend 6 months to a year learning.
While every tailor's methods will differ a little, the basics remain more or less similar to this:
It requires 3 types of cloth; body canvas, chest (or horsehair) canvas and domette.
As I mentioned, each tailor will cut and arrange these slightly differently but once they will all "pad" them. This is a series of reasonably large stitches designed to create a curve in the chest to accommodate the pecs. As a tailor sews, he will curve the canvases around his thigh, the stitches ensure that curve will remain even after years of wear. A smaller version of this stitch is used on the collar as well.
One of the best things about a bespoke jacket is the comfort factor. As the canvases are cut and shaped, they are built up at the collar bone. By creating length here, the coat will come away a little rather than rubbing against the bone.
The forepart of the coat is then laid on top of the canvas and attached using 4 rows of carefully placed stitching. A bridle is used to draw in the break line of the lapel to stop any gaping that can occur here, particularly on men with larger chests. The "pad" stitch is then used at the lapels to make sure they roll nicely and sit flatly on the chest.
Any inlay (the extra cloth added for adjustments) is then neatly trimmed away. To make sure the seams are as flat as possible, the canvas is cut down to just over 1 seam smaller than the coat. Linen tape is lashed onto the canvas at the edge. Linen is much thinner than the canvas so will produce neater seams. The pocket bags are attached to the canvas to give them strength over years of wear and tear.
Finally, once the sleeves have been set in, sleevehead roll (which is used to make the top of the sleeves smooth) will be sewn through the canvas and sleeve seam further securing the canvas.
All of the padding, lashing, securing and shaping serves to ensure your coat continues to look good for years.