The Azure Damselfly can be distinguised from the similar looking Common Blue Damselfly (Enallagma cyathigerum) by having more black on its back and by having two little black stripes on the side of its thorax.
The Common Blue Damselfly has only one little black stripe on the side of its thorax.

Also, the male Azure Damselfly (Coenagrion puella) resembles the male Variable Damselfly (Coenagrion pulchellum).
However, the male Variable Damselfly has a distinctive "wine glass" marking on the second segment of the abdomen.
This is a black U-shaped mark with a black line joining the segment's narrow terminal black band.
This distinguishes it from the Azure Damselfly which has the U-shape but no line connecting it to the terminal band.

1st photo: Zanderij Crailo, Hilversum, The Netherlands, 22 June 2013
2nd photo: De Snip, Hilversum, The Netherlands, 18 June 2013
3rd & 4th photo: Botanical Garden, Utrecht, The Netherlands, 7 July 2011