Psalm Tones
There are various tones used for chanting the psalms.
Archdiocese of Omaha Psalm Tones
Chanting the Psalms by Fr. Samuel F. Weber (earlier version) contains an explanation on how to apply tones to the psalm texts. It has a list of tones, some of which overlap with the tones found in the Mundelein Psalter.
Here are descriptions of each of the eight Gregorian modes:
One question may be when are each each of the tones used? Are certain psalms matched with certain tones? The short answer is there is no set rule. The same psalm will be found in the Graduale Romanum with different tones, tones are not specific to a certain psalm. Also you find all the tones used in every season, so there is no specific rule there.
In general, you do find certain modes that are predominant in certain seasons:
In praying the Liturgy of the Hours, one could apply what the Mundelein Psalter uses for the distribution of modes, taken from here.
An interesting discussion on all this can be found here.