How do you calculate declination?

A star's declination and its latitude are the same thing.

A star's declination, δ, can be found like this:

δ = ArcSin { sin φ sin E + cos φ cos E cos A }

where

φ = the observer's latitude

E = the star's elevation

A = the star's azimuth

Declination is the measurement of the star's position above or below the celestial equator. As previously stated, cannot be calculated. It can be measured easily enough. Setting circles on a properly set up telescope, even a stick and protractor with weighted string and where in the sky the celestial equator is at that time.

The position of a star on the celestial sphere is marked with 'Right Ascension' (from 0 to 24 hours) and 'Declination' (from -90 to + 90 degrees).

The position of the star above the horizon is marked with 'Azimuth' (along the horizon from 0 to 360 degrees) and 'Altitude' (height above horizon, from 0 to 90 degrees).

If you need to calculate above from the first system to the second system, or opposite, then you need the 'latitude' of the observer on earth (from -90 South to +90 North) and his 'time'. (Think of the rotating earth in a day time.) These are mathematical formulas with lots of 'cos' and 'sin', but it can.

Edit: you can also wait until the star is highest, which is always in south, and then the declination simply is:

AltitudeStar - 90 + LatitudeObserver

Coldfieldgirl has described the different systems well. The conversion calculation from Alt/Az to RA/Dec is a tad tricky (understatement!) - but there's a calculator and a rather scary text description in the supplied links.

Warning - I haven't thoroughly examined either for errors!

Would I use them? No way! I just use a decent planetarium program like Stellarium...