The idea of the chuma, or stringency, in Orthodox Judaism looms large. G-d appointed the Jewish people His holy nation and gave us His Law, but individuals have always strived to do more. There are actually a few different categories of chumra. Scholars have written about the subject, but in the form of manuals for getting closer to G-d. I'm just describing people's actions as they appear.
Ever since the disbanding of the Sanhedrin (central government of rabbinic judaism), religious legislation has been somewhat up in the air. This situation leads to the most fashionable type of chumra, the attempt to
satisfy all the conflicting decisions made by a group of different decisors.
The group of decisors is often the rishonim (medieval scholars), and satisfying their decisions inspires many of the famous "Brisker Chumros". These chumros are more compelling when the issue is a matter of Biblical law rather than Rabbinic, but often the provenance itself is also in question. It is literally impossible to always satisfy
all of the decisions of all the rishonim. [To be written: rashi and rabbenu tam tefillin example,
rov example]
As we consider later decisions, it becomes increasingly less clear whether a given position is a chumra or actual halacha. At least for opinions of the rishonim, we usually have a idea of the halacha from the Shulchan Aruch. After that, we often have many different strong positions and no agreed upon principle for deciding amongst them. These questions are usually either less fundamental, or else they deal with technology and new situations.
A subcategory of satisfying
all decisions is to
observe authoritative decisions that are mostly disregarded. This disregard may be explained away by later authorities, or it may be mysterious. Either way, we are reluctant to suppose that the majority of observant jews are violating halacha, so these decisions take on the status of chumra. An example is Ezra's decree that after seminal emission, men should immerse in the Mikveh before learning or praying.
Another major category of chumra is that of
ascetic practices. In general, the torah encourages--and perhaps requires--embracing life, but historically scholars have valued greater restraint. Maimonides famously praises the ideal of moderation, but his view of moderation looks harsh by today's standards. An example is fasting most Mondays and Thursdays.
Excruciating attention to detail is another major form of chumra, which some consider to be part of normative halacha. The classic example is that halacha specifies the proper order for tying your shoes. This trend goes back to the Talmud, which wrangles over the exact right place to first break open a loaf of bread. People may try to be meticulous in interpersonal relations as well (for example, not telling a companion at a restaurant that you dislike your meal, lest the waiter overhear and get hurt feelings).
Many of these chumras evoke awareness of G-d and godliness in every aspect of one's life, even the most picayune. It could be that accomplishing this awareness is more important than the actual details themselves.
Mysticism informs mainstream halacha, but it also forms a rich body of chumra. Presumably this category need not be limited to the traditional literature, but may include fringe texts, and even personal research into hidden worlds.
Much of civil law is about compromise, where neither party to a dispute is actually at fault. The Talmud describes "chasidus" as the habit of
forgoing legal advantage in many such situations, and exceeding one's pecuniary obligation. [Source in bava kamma]
The last category of chumra is a biggy:
do more good. This means strengthening the pillars of the world: learning more torah, giving effort and money to those in need, and praying more meaningfully.