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With the sample MWE below, TikZ generates the following diagram:

1. Drawing an arc

documentclassarticle
usepackagetikz
begindocument
begintikzpicture
 coordinate[label=below:$a$] (a) at (0,0);

 draw (a) arc(0:180:2);
endtikzpicture
enddocument 

2. Drawing a circle

documentclassarticle
usepackagetikz
begindocument
begintikzpicture
 coordinate[label=below:$a$] (a) at (0,0);
 coordinate[label=below:$b$] (b) at (5,5);

 draw (a) arc(0:180:2);
 draw (b) circle [radius=2cm];
endtikzpicture
enddocument 

With draw <coordinate> circle command, TikZ draws a circle, with the <coordinate> at its center.

With draw <coordinate> arc command, TikZ draws an arc, with the <coordinate> as one of its endpoints.

I wish to understand, why this difference of behavior and how is TikZ drawing the arc with draw (a) arc(0:180:2);.

Why does TikZ does not use (a) as its center point while drawing an arc similar to drawing a circle.

How does TikZ actually draw the arc with point a.

What is the reference and center point, when TikZ draws an arc.

Here is an attempt to use an animation to explain step by step what's going on. I use the non-deprecated syntax, as suggested by Joule V.

documentclassbeamer
usepackagetikz
usetikzlibraryoverlay-beamer-styles
begindocument
beginframe[t]
frametitleThe arc construction
textbackslashtextttdraw (P) arc[start angle=$alpha$,end 
angle=$beta$,radius=$r$];

centering
begintikzpicture
 node[circle,fill,inner sep=1pt,label=right:$P=(x,y)$] (P);
 draw[dashed,visible on=<2-3>] (P) ++ (210:2) coordinate (O) circle[radius=2cm];
 draw[visible on=<2>] (O) + (1,0) arc[start angle=00,end angle=30,radius=1cm] 
 node[midway,right] $alpha$;
 draw[visible on=<2>] (P) -- (O) -- ++(2,0) node[midway,below]$r$;
 draw[visible on=<3-4>,thick] (P) arc[start angle=30,end angle=135,radius=2cm]
 coordinate (Q);
 draw[visible on=<3>] (O) + (1,0) arc[start angle=00,end angle=135,radius=1cm] 
 node[midway,below] $beta$ (Q) -- (O) -- ++(2,0) ;
 path (O) -- ++ (0,2.5); % only for the bounding box
endtikzpicture
beginitemize
 item<+-> Consider a point $P=(x,y)$.
 item<+-> Imagine now a circle of radius $r$ for which $P$ sits at the angle
 $alpha$.
 item<+-> The arc runs along this circle between the angles $alpha$ and
 $beta$.
enditemize
endframe
enddocument

Short answer: arc has nothing to do with circle. circle is drawn with four curves, arc is drawn with one curve, that is all.

Let's start with tikz.code.tex. In lines 3673–3706, there is the definition of tikz@do@arc, based on pgfpatharc, which is defined in pgfcorepathconstruct.code.tex. Circles and ellipses are defined in the same way.

In pgfcorepathconstruct.code.tex, pgfpatharc is defined as a curve with a starting point and an ending point, based on pgf@nlt@curveto (line 401).

Meanwhile, you can see that any kind of curves, even circles, are defined based on pgf@nlt@curveto. Circles and ellipses is defined in pgfpathellipse as a set of four connected curves: line 926, line 947, line 968, and line 989 — each of these curves has its own pgf@nlt@curveto.

That is why the arc does not take its coordinate as the center, but a starting point.

A PSTricks solution just for fun.

In PSTricks, drawing circular arcs as well as elliptical arcs is very intuitive as follows.

documentclass[pstricks,border=12pt,12pt]standalone
begindocument
beginpspicture[dimen=m](4,4)
 pscircle[linestyle=dashed](2,2)2
 psellipse[linestyle=dashed](2,2)(2,1)
 psarc[linecolor=red](2,2)24590
 psellipticarc[linecolor=red,correctAngle=true](2,2)(2,1)4590
endpspicture
enddocument