A graphical method
Sometimes a graphical method may be easier to understand and remember.
One year is the time required for the planet to complete one full orbit around its primary. You decide how long a year is; it could be shorter than an Earth year, it could be longer; but if the star is similar to our Sun, and the planet is supposed to be habitable for life as we know it, it cannot be all that much shorter or all that much longer.
A day can be reckoned in two ways.
The simplest way is the time required for the planet to complete a full rotation around its axis; this is called a sidereal day, because it is the time between two culminations of any given star. ("Sidera" means stars in Latin.)
You choose the length of the sidereal day. It can shorter than an Earth day, it can be longer. There is no relationship between the length of the year and the length of the sidereal day.
The more complicated, but more useful way is to reckon the time between two culminations of the planet's sun, that is, the time from one noon to the next; this is called a solar day, and it is a bit longer than a sidereal day, and can be calculated as the duration of one sidereal day plus a fraction of the sidereal day equal to the ratio between the sideral day and the year.
For Earth, a sidereal day is 23 hours 56 minutes, and an average solar day is 24 hours. (The current definition of the second was chosen very carefully, so that the mean solar day computed for 1 January 1900 is almost exactly 24 hours.)
(The solar day is longer than the sidereal day because by the time the planet has completed one rotation around the axis it has also advanced a little bit on its orbit, and it must rotate a little bit more to bring the sun in the same position.)
(Of course, if the planet rotates in the opposite direction that its revolution around the primary, the solar day will be shorter than the sidereal day, with the same amount. Most planets don't do this -- they rotate around the axis and revolve around the primary in the same direction.)
As the planet revolves around its primary there are four important points on the orbit;
At one point, the axis of rotation appears to be tilted towards the primary at a maximum, equal to the obliquity. This is the northern solstice, which is the summer solstice for people in the northern hemisphere. At the northern solstice, at all places on the planet north of the northern polar circle, that is, the northern parallel of 90° less the axial tilt, the sun doesn't set; and at all places on the planet south of the southern polar circle, that is, the southern parallel of 90° less the axial tilt, the sun doesn't rise.
Then comes a point where the axis of rotation is perpendicular to the radius of the orbit; this is an equinox. At equinoxes, days and nights are equal at all latitudes.
Then comes a point where the axis of rotation appears to be tilted away from primary at a maximum, equal to the obliquity. This is the southern solstice, which is the winter solstice for people in the northern hemisphere. At the southern solstice, at all places on the planet north of the northern polar circle, that is, the northern parallel of 90° less the axial tilt, the sun doesn't rise; and at all places on the planet south of the southern polar circle, that is, the southern parallel of 90° less the axial tilt, the sun doesn't set.
Finally, a second point where the axis of rotation is perpendicular to the radius of the orbit; this is an equinox. At equinoxes, days and nights are equal at all latitudes. Then the cycle repeats.
Assumming that the planet has a circular or almost circular orbit around its primary, the four points (two solstices and two equinoxes) are almost equally spaced within the year.
What you want to do is compute the duration of daylight for a given latitude at the northern solstice; then you can estimate the duration of daylight for that latitude at any time in the year.
How to compute the duration of day and night at summer or winter solstice for a given latitude using a graphical method. Own work, available on Flickr under the Creative Commons Attribution 2.0 Generic license.
Draw the planet tilted towards the Sun.
Draw the equator; notice that on the equator the days and nights are of equal length at all times.
Draw the polar circles as lines parallel to the equator starting from the topmost and bottommost points on the planet.
Draw the terminator, that is, the line separating day from night. Note the position of the terminator with respect to the lines representing the polar circles.
With a protractor, identify your parallel of interest. In the picture, the parallel of interest is at 30°.
Now measure how much of that parallel is in the illuminated part of the planet, and how much is in the shadow.