A Chumra is the incorporation of a legitimate stringency that is not the Halachic ruling into one's behavior . By legitimate stringency I mean that there often exists in Psak Halacha differing Shitos as to what the appropriate performance of a Mitzva should take. The Shulchan Aruch’s ruling will usually be lenient in Halacha based on the majority of Rishonim who are lenient (Mekil) vs. the minority who are stringent (Machmir). But there still remains the possibility that the more stringent ruling by other Rishonim might be the correct one. So adopting the more stringent view would be adopting a Chumra.
An example would be the use of an Eruv in a metropolitan area.
To explain this Halacha briefly, a Reshus HaRabim D’Oraisa (a biblically defined public area) cannot in any way be turned into a Reshus HaYachid (private domain). A Reshus HaRabim D’Oraisa is defined as any area that has streets sixteen Amos (Cubits) wide and a population of at least 600,000 people. If, however, an area contains less then 600,000 people making it a biblical Reshus HaYachid (Karmelis) then carrying is prohibited only rabbinically. Our sages mandated that since it is similar looking to a Reshus HaRabim D’Oraisa it is then deemed a Reshus HaRabim D’Rabbanan and they did not allow carrying in it. A Karmelis can be adjusted to become a rabbinical Reshus HaYachid that would allow carrying.
This is done by means of an Eruv. An Eruv is in essence an enclosure surrounding a Karmelis on all four sides. The exact parameters of an Eruv are beyond the scope of this essay. But, clearly, mainstream Jewish law allows the use of the Eruv so that one can carry objects in public areas.
So, getting back to the subject at hand, the Chumra would be, for those whose view follows that of the Rambam, a minority view that a Reshus HaRabim D’Oraisa requires only that streets to 16 Amos wide and does not require 600,000 residents. According to the Rambam a Karmelis is a Reshus HaRabim D’Oraisa and cannot be made into Reshus HaYachid by means of an Eruv. Carrying would be prohibited biblically. So, in any given city where there are typical streets, no Eruv would help no matter how small the population.
So not using an Eruv in metropolitan areas, no matter how many stringencies are built into it is indeed a legitimate Chumra ...and this is the position of The House of Brisk. They consider all metropolitan Eruvin to be useless.
However, if there is no legitimate view that is stringent on specific performance of a Mitzvah, then being stringent is an unnecessary exercise that is self defined and has no purpose.
Minhag or custom, though sometimes very nice, should not fall into the category of Chumra unless it has a source in Rishonim. For example, wearing a black hat during prayer or eating Matza Balls in one's is not a Chumra.
Levels of observance also do not fall into the category of Chumra. If the sages mandate three forms of observance of a particular mitzvah, than there are three levels of Hidur, meaning preferable or higher form of observance, not three levels ranging from Kula to Chumra. Lighting the Hanukkah Menorah with an additional candle every night is a Hidur over lighting one candle per night. It is not a Chumra.
It is interesting to note that one man's Kula is another man's Chumra. Rabbi Eliahu Soloveichik told me that Brisk is very stringent on Brachos (blessings). They do NOT make the Shehechyanu blessing on purchases of new clothes. The Chumra is...not to make an unnecessary Bracha which if said violates the prohibition of saying God’s name in vein.
