Diminished chords and the blues
To make a C diminished seventh chord, start on C. Go up a minor third (three frets) to Eb. Then go up another minor third to Gb, and another minor third to A. If you continue up by another minor third, you'll arrive back at C. The chord symbol is C°7 or Cdim7.
Here are three fingerings for C°7. I=index, M=middle, R=ring, P=pinkie.

Root on low E string

 

6

7

8

9

  notes

E

x x x x  

B

-- I -- -- barre
with
index

G

-- I R --

D

-- I -- --

A

x x x x  

E

-- -- M -- root

Root on high E string

 

6

7

8

9

  notes

E

-- -- P -- root

B

-- M -- --  

G

-- -- R --  

D

-- I -- --  

A

x x x x  

E

x x x x  

Root on A string

 

1

2

3

4

5

  notes

E

x x x x x  

B

-- I -- P -- barre
with
index

G

-- I -- -- --

D

-- I -- R --

A

-- -- M -- -- root

E

x x x x x  
If you move a C°7 up a minor third (three frets), you'll get an Eb°7, but this chord is also C°7 - the same four pitches, but in a different order. Move it up another minor third, and another, and you'll get the same result (go ahead, try it - you've just played a cliche common from old horror movie scores.) All four of the chords below are identical and completely interchangeable.
C°7 = Eb°7
D#°7
= Gb°7
F#°7
= A°7
The same is true of the other diminished chords:
Db°7
C#°7
= E°7 = G°7 = Bb°7
A#°7
  and   D°7 = F°7 = Ab°7
G#°7
= B°7
If you think about it, there are really only three possible diminished chords total.
So what do you use these things for?
As you've probably discovered by now, diminished chords sound pretty peculiar when played by themselves. They sound best when inserted in between two other chords as a transition or an accent. They were a very common songwriting device in the pre-rock era and are still a staple of jazz.

Diminished chords have a special place in the blues. First of all, the chord itself has a bluesy feel because it contains a flatted fifth. Secondly, there are some classic blues licks that incorporate diminished chords into turnarounds (short progressions that start and end on the same chord.) Some examples in the key of C follow. (You can, of course, transpose these to any other key.)
Robert Johnson intro - roots on high E string

C7

  8 9 10 11 12

E

-- -- -- -- p

B

-- -- -- m --

G

-- -- -- -- r

D

-- -- i -- --

A

x x x x x

E

x x x x x

Eb°7

  8 9 10 11 12

E

-- -- -- p --

B

-- -- m -- --

G

-- -- -- r --

D

-- -- i -- --

A

x x x x x

E

x x x x x

D°7

  8 9 10 11 12

E

-- -- p -- --

B

-- m -- -- --

G

-- -- r -- --

D

-- i -- -- --

A

x x x x x

E

x x x x x

C7

  8 9 10 11 12

E

i -- -- -- --

B

i -- -- (p) --

G

i m -- -- --

D

i -- -- -- --

A

x x x x x

E

x x x x x
A common blues lick is to play these fingerings on only the G and high E strings:

C7

  8 9 10 11 12

E

-- -- -- -- p

B

x x x x x

G

-- -- -- -- r

D

x x x x x

A

x x x x x

E

x x x x x

Eb°7

  8 9 10 11 12

E

-- -- -- p --

B

x x x x x

G

-- -- -- r --

D

x x x x x

A

x x x x x

E

x x x x x

D°7

  8 9 10 11 12

E

-- -- p -- --

B

x x x x x

G

-- -- r -- --

D

x x x x x

A

x x x x x

E

x x x x x

C7

  8 9 10 11 12

E

i -- -- -- --

B

x x x x x

G

-- m -- -- --

D

x x x x x

A

x x x x x

E

x x x x x
Also, try playing either version backwards.
Robert Johnson intro - roots on A string

C7

 

1

2

3

4

5

E

x x x x x

B

-- -- i -- p

G

-- -- i -- --

D

-- -- i -- r

A

-- -- i -- --

E

x x x x x

C°7

 

1

2

3

4

5

E

x x x x x

B

-- i -- p --

G

-- i -- -- --

D

-- i -- r --

A

-- -- m -- --

E

x x x x x

B°7

 

1

2

3

4

5

E

x x x x x

B

i -- p -- --

G

i -- -- -- --

D

i -- r -- --

A

-- m -- -- --

E

x x x x x

C7

 

1

2

3

4

5

E

x x x x x

B

i -- -- -- --

G

-- -- p -- --

D

-- m -- -- --

A

-- -- r -- --

E

x x x x x
To do the single-string lick, play these fingerings on only the D and B strings:

C7

 

1

2

3

4

5

E

x x x x x

B

-- -- -- -- p

G

x x x x x

D

-- -- -- -- r

A

x x x x x

E

x x x x x

C°7

 

1

2

3

4

5

E

x x x x x

B

-- -- -- p --

G

x x x x x

D

-- -- -- r --

A

x x x x x

E

x x x x x

B°7

 

1

2

3

4

5

E

x x x x x

B

-- -- p -- --

G

x x x x x

D

-- -- r -- --

A

x x x x x

E

x x x x x

C7

 

1

2

3

4

5

E

x x x x x

B

i -- -- -- --

G

x x x x x

D

-- m -- -- --

A

x x x x x

E

x x x x x
And again, try it backwards.
Don't Get Around Much Anymore lick - roots on high E string

C7

  8 9 10 11 12

E

i -- -- -- --

B

i -- -- -- --

G

i m -- -- --

D

i -- (r) -- --

A

x x x x x

E

x x x x x

D-7

  8 9 10 11 12

E

-- -- i -- --

B

-- -- i -- --

G

-- -- i -- --

D

-- -- i -- --

A

x x x x x

E

x x x x x

D#°7

  8 9 10 11 12

E

-- -- -- p --

B

-- -- m -- --

G

-- -- -- r --

D

-- -- i -- --

A

x x x x x

E

x x x x x

C7

  8 9 10 11 12

E

-- -- -- -- p

B

-- -- -- m --

G

-- -- -- -- r

D

-- -- i -- --

A

x x x x x

E

x x x x x
Don't Get Around Much Anymore lick - roots on A string

C7

 

3

4

5

6 7 8

E

x x x x x x

B

i -- p -- -- --

G

i -- -- -- -- --

D

i -- r -- -- --

A

i -- -- -- -- --

E

x x x x x x

D-7

 

3

4

5

6 7 8

E

x x x x x x

B

-- -- i m -- --

G

-- -- i -- -- --

D

-- -- i -- r --

A

-- -- i -- -- --

E

x x x x x x

D#°7

 

3

4

5

6 7 8

E

x x x x x x

B

-- -- i -- p --

G

-- -- i -- -- --

D

-- -- i -- r --

A

-- -- -- m -- --

E

x x x x x x

C7/E

 

3

4

5

6 7 8

E

x x x x x x

B

-- -- i -- -- p

G

-- -- i -- -- --

D

-- -- i -- -- r

A

-- -- -- -- m --

E

x x x x x x
Any of these turnarounds can be used in place of C7 - for example, in the first four bars of blues in C, or in any rock or country tune that stays on C7 for a few bars.
Boilerplate blues intro

C7

  7 8 9 10

E

x x x x

B

-- i -- --

G

-- i m --

D

-- i -- --

A

-- i -- R

E

-- i -- --

C9/E

  7 8 9 10

E

x x x x

B

-- P -- --

G

m -- -- --

D

-- R -- --

A

i -- -- --

E

x x x x

F7

  7 8 9 10

E

x x x x

B

-- i -- p

G

-- i -- --

D

-- i -- r

A

-- i -- --

E

x x x x

F#°7

  7 8 9 10 11

E

x x x x x

B

-- i -- p --

G

-- i -- -- --

D

-- i -- r --

A

-- i m -- --

E

x x x x x

G7#9

  8 9 10 11

E

x x x x

B

-- -- -- p

G

-- -- r --

D

-- i -- --

A

-- -- m --

E

x x x x
Pay special attention to the bass notes; they're the heart and soul of this lick. You can use any voicing of G7; the one above is my favorite for this situation.
For an ending, add this or something similar:

Db9

  1

2

3

4

5

E

-- -- -- r --

B

-- -- -- r --

G

-- -- -- r --

D

-- -- i -- --

A

-- -- -- m --

E

x x x x x

C9

  1

2

3

4

5

E

-- -- r -- --

B

-- -- r -- --

G

-- -- r -- --

D

-- i -- -- --

A

-- -- m -- --

E

x x x x x
Soloing on diminished chords
In the blues/rock/country context, the C blues scale will work fine over any of the progressions above. In a jazz setting, you can try the C diminished scale on a C°7, an exotic and mysterious-sounding collection of notes for sure:
C -- D Eb
D#
-- F Gb
F#
-- Ab
G#
A -- B

Experiment and use your ears!