Círdan the Shipwright's Beard
Based on Barrow-Downs Forum material compiled by Legolas
Any errors of spelling or grammar have been corrected silently.

"I've been wondering this for the longest time: Does Círdan have a beard? I know I've read somewhere that elves don't have facial hair, but then what's the explaination for Círdan?" (-Lothiriel Silmarien)

"Círdan is an elf, right? Then why is he is described as appearing old and with a beard in The Return of the King?" (-Rust Ayokae)

Elves' lives are divided into three periods. Some elves grow beards during the third division; there are rare exceptions where an elf will have a beard in the second. Círdan is presumed to be quite old, in this third period at the time of the War of the Ring.*

"Círdan does have a beard. It is considered part of the aging process elves go through. There are three levels and Círdan is probably in the third one since he has been alive so long. Staying in Middle-Earth didn't help with his aging either.
Here is where it says he has a beard in The Lord of the Rings:

As they came to the gates Círdan the Shipwright came forth to greet them. Very tall he was, and his beard was long, and he was grey and old, save that his eyes were keen as stars; and he looked at them and bowed, and said: 'All is now ready.'

Círdan was not the only elf to have a beard either. This is from the Vinyar Tengwar:

A note elsewhere in the papers associated with this essay reads: "Elves did not have beards until they entered their third cycle of life. Nerdanel's father [cf. XII:365-66 n.61] was exceptional, being only early in his second."

So you see elves could grow beard, they just have to be really old." (-Durelen)

"It is hard for me not to touch the matter of Círdan, as some may know... however, we also have a pretty bearded Beleg in the old painting by the Prof first entitled 'Beleg finds Gwindor [by then still Flindling, actually] in Taur-Nu-Fuin', later entitled 'Mirkwood'." (-Sharkű)

Primary sources:
Vinyar Tengwar
The Return of the King

Forum resources:
On Círdan's Beard I*
On Círdan's Beard II
On Elven Aging