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I want to draw the following diagram in LaTeX:

It's quite easy with tikz-cd:

documentclassarticle
usepackagetikz-cd
begindocument
begintikzcd
1 ar[r,"a"] ar[rr,bend right,"c"'] & 2 ar[r,"b"] & 3
endtikzcd
enddocument

But, in my humble opinion, I'd vote for picture-mode, for its far more user-friendly syntax:

documentclassarticle
usepackagepict2e
begindocument
setlengthunitlength1cm
beginpicture(2.1,0.75)(0,-0.45)
put(0,0)1
 put(0.2,0.1)vector(1,0)0.75
 put(0.5,0.15)footnotesize$a$
put(1,0)2
 put(1.2,0.1)vector(1,0)0.75
 put(1.5,0.15)footnotesize$b$
put(2,0)3
 qbezier(0.2,-0.05)(1.05,-0.7)(1.9,-0.05)
 put(1.9,-0.05)vector(4,3)0.0001
 put(1.0,-0.55)footnotesize$c$
endpicture
enddocument

Update: tikz-cd solution for the second diagram:

documentclassarticle
usepackagetikz-cd
begindocument
begintikzcd
 1 ar[r,"a_1"]
 ar[rrrrr,bend right=50,looseness=0.3,"b"']
& 2 ar[r,"a_2"]
& 3 ar[r,"a_3"]
& cdots
 ar[r,"a_n-2"] % This arrow isn't in the picture, but I chose to add it. Feel free to remove.
& n-1 ar[r,"a_n-1"]
& n
endtikzcd
enddocument

For the sake of completeness, here it is a plain TikZ approach

documentclassarticle
usepackagetikz
usetikzlibrarypositioning
begindocument
 begintikzpicture[node distance = 2cm, thick]% 
 node (1) 1;
 node (2) [right=of 1] 2;
 node (3) [right=of 2] 3;
 draw[->] (1) -- node [midway,above] a (2);
 draw[->] (1) to [bend right] node [midway,below]c (3);
 draw[->] (2) -- node [midway,above] b (3);
 endtikzpicture%
enddocument

documentclassarticle
usepackagetikz
usetikzlibrarypositioning
begindocument
 begintikzpicture[node distance = 2cm, thick]% 
 node (1) 1;
 node (2) [right=of 1] 2;
 node (3) [right=of 2] 3;
 node (dots) [right=of 3] $cdots$;
 node (n1) [right=of dots] $n-1$;
 node (n) [right=of n1] $n$;
 %
 draw[->] (1) -- node [midway,above] $a_1$ (2);
 draw[->] (1) to [bend right=15cm] node [midway,below]b (n);
 draw[->] (2) -- node [midway,above] $a_2$ (3);
 draw[->] (3) -- node [midway,above] $a_3$ (dots);
 draw[->] (dots) -- node [midway,above] $a_n-2$ (n1);
 draw[->] (n1) -- node [midway,above] $a_n-1$ (n);
 endtikzpicture%
enddocument

A pstrickssolution: the three numbers are defined as nodes, and we link them by node connections.

documentclass[12pt]article
usepackagepst-node, auto-pst-pdf

begindocument

[
 Rnode11 hspace3em Rnode22 hspace3em Rnode33
pssetlinewidth=0.6pt, arrows=->, arrowinset=0.12, nodesep=0.6ex, labelsep=2pt, shortput=nab
everypsboxscriptstyle
ncline12^ancline23^b
ncarc[arcangle=-30]13_c
]

enddocument 

With tikz, it is possible to use the chain concept of the graphs library.

chains are a series of nodes whose syntax is abbreviated compared to their normal syntax. A chain ends with a semicolon. The nodes and their text are identical by default as in the first figure. Nodes and text are the number 1, 2 and 3.

This first figure is composed of 2 chains: the first 1->2->3 and the second which goes directly from 1 to 3.

To add text to the arrow (that is called edge by tikz), simply place the option edge label or edge label' depending on whether the text is above or below the arrow. In reality, this places the text to the left or right when walking in the direction of the chain.

In the second example, to avoid having to draw an arrow, we write -!- instead of -> (see p 265 of the manual).
To make the node text different from its name, simply separate the name from the text with a slash. For example : 5/$n-1$ means that node 5 displays the n-1 text.

documentclass[a4paper,12pt]article
usepackagetikz
usetikzlibrarygraphs
begindocument

tikz graph[grow right=30mm] 
1 ->[edge label=a] 2 ->[edge label=b] 3;
1 -> [bend right,edge label'=c] 3;
;


tikz graph[grow right=20mm] 
1 ->[edge label=$a_1$] 2 ->[edge label=$a_2$] 3->[edge label=$a_3$] 4/$cdots$ -!- 5/$n-1$->[edge label=$a_n-1$] n ;
1 -> [bend right,edge label'=b] n;
;

enddocument

Edit:

Using the quotes library, the syntax for writing labels is even shorter:
"a" instead of edge label=a
"c"' instead of edge label' =c
You can customize the colors as below:

documentclass[a4paper,12pt]article
usepackagetikz
usetikzlibrarygraphs,quotes
begindocument

tikz[thick,gray] graph[grow right=30mm] 
1 ->["a"] 2 ->["b"] 3;
1 -> [bend right,"c"'] 3;
;

General case:
smallskip

tikz[thick,every edge/.style=red,draw,every node/.style=blue] graph[grow right=20mm,edge quotes=black,auto] 
1 ->["$a_1$"] 2 ->["$a_2$"] 3->["$a_3$"] 4/$cdots$ -!- 5/$n-1$->["$a_n-1$"] n ;
1 -> [bend right,"b"'] n;
;

enddocument

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