Kashrut is the system of Jewish dietary laws. Food permitted under Kashrut is called kosher.
There are only a few major rules of kashrut, but they have many applications.
During Passover (Pesach) no leavened bread may be consumed. The rules on this are rather strict and any flour that has been in contact with water for over 18 minutes is considered leavened and mustn't be used during Pesach. In addition, Ashkenazi Jews traditionally refrain from other products known as kitniyot, the exact definition of which depends on which rabbi you ask, but which include rice and some legumes.
Products that are kosher for Pesach are often specially certified saying so.
While not as difficult as it once was, finding kosher food while traveling can still be a challenge.
One possibility is just to bring your own kosher food from home. While this may seem to be the easiest way, it is often not. For even modestly long journeys, it may be difficult to carry the requisite amount of food. It is often impractical to refrigerate perishables (although food can be kept frozen in an airplane, as long as the plane is not delayed on the ground for too long), and while non-perishable food items will travel more easily, you will need extra space to pack them. It is sometimes illegal to move fruit and the like across country borders. Kosher self-heating meal kits are an increasingly popular food item for Jewish travelers, and while they are something to consider, they are frequently banned from aeroplanes. Good non-perishable, calorie-dense, non-fragile foods to bring include dried fruit, nuts, granola bars, rice cakes, and canned goods.
Food sold in grocery stores is usually packaged, and is sometimes kosher. You will need to look for the kosher certification label or seal on the package to verify that the food item is actually kosher. While at least some kosher packaged food is readily available in most grocery stores in North America and Europe, it may be more difficult to find elsewhere, and will almost always be imported from another country.
Some synagogues and Jewish outreach centers (such as Chabad Houses) offer packaged or frozen food items, prepared foods, and even a selection of meats for sale for kosher travelers. You will need to contact the synagogue or outreach center directly to find out about what is available, how to purchase, and how to obtain the food; in many cases you may have to reserve the food in advance. This can be especially helpful in cities or countries where a Jewish community presence is limited. If visiting around a major Jewish holiday or on the Sabbath, you may also be able to partake in a full meal at the synagogue or outreach center, usually at no charge (though a donation to the center may be requested).
Most of the rules of kashrut involve meat, and there is some overlap with Islamic dietary laws, so depending on your level of observance, you might feel comfortable just going for a vegetarian or halal option when certified kosher food isn't available. Of course, many observant Jews won't consider this an acceptable option. In many places foods which are vegan or vegetarian will also carry additional "halal" or "kosher" labels, but this mostly depends on whether the extra cost to get certified can be recouped by increased sales.
Kosher food was one of the first "special foods" to be introduced on commercial airplanes and it is widely available today. You will usually have to specify a kosher meal in advance to get one. Israeli flag carrier El Al only offers kosher meals on all flights, a tradition that dates back to the founding of the airline. When departing from airports that do not have a commercial Kosher kitchen, be prepared to receive a tray of fruit or granola bar instead of a full meal, and plan in advance for alternative meal options.
Israel is the most welcoming country in the world for kosher travelers. Thousands of kosher restaurants, dining establishments, and grocery stores can be found throughout the country, most notably in Jerusalem and Tel Aviv. A novelty for most Jewish travelers are Israel's all-kosher adaptations of famous American fast-food chains, including McDonald's and KFC.
Parts of the United States are on par with Israel in the number of kosher dining options and establishments that are available. Major cities such as New York, Baltimore, Miami, Chicago, and Los Angeles have a particularly notable variety of kosher restaurants in operation, and most, if not all, sizable metropolitan areas will have at least one kosher establishment, often a certified bakery or deli counter at a supermarket. You will also find at least some kosher-certified packaged foods or non-perishable items can typically be found in all but the most remote grocery stores of the country. Many major brands found in grocery stores carry kosher certification from the Orthodox Union, signified by the circled-U hechsher symbol, Ⓤ, on food packaging. As even many non-Jewish Americans perceive kosher products to be of higher quality, products which are kosher "by default" (e.g. vegan products) are usually also labeled as kosher as the small extra cost of getting certified is well worth the extra sales.
For travellers touring U.S. universities or studying in the United States, university cafeterias sometimes have a designated kosher station, and even if they don't, there's often some kind of program on campus to make kosher meals available.
Toronto and Montreal in Canada both offer a good number of kosher restaurants as well. Other major cities like Vancouver and Calgary will also have at least one or two kosher establishments.
In Mexico, there is a significant Jewish community in Mexico City that operates several kosher eateries. There is also a Jewish presence (and Chabad houses) in the Yucatan area.
England (London in particular) has a decent variety of kosher establishments, mostly situated in the North London neighborhoods of Golders Green, Hendon, Edgware, and Stamford Hill. Paris also has a large Jewish community, primarily descended from Sephardic and Mizrahi Jewish refugees from France's North African and Middle Eastern former colonies. In most of continental Europe, kosher dining options are often few and far between and where they exist, they sadly are sometimes target of anti-Jewish violence, so additional safety precautions may have to be taken (e.g. a police guard or metal detectors at the entrance). In order to find the next kosher store, inquiring at the local synagogue is always a good idea. One of the European countries where a surprisingly high number of Jews live (and, hence, kosher dining options exist) is Germany. As many German Jews are or are descended from recent emigrants from the former Soviet Union, kosher food options may sometimes be found in "Russian" stores as well. Kosher restaurants also exist, but they are rather few and far between.
Very few kosher establishments are located in Asian countries (with the obvious exception of Israel). Most likely, you will need to bring/find your own food or contact a local Jewish organization at your destination for assistance. In some areas with no kosher butchers or restaurants, the local Chabad house provides kosher meals for Jewish travellers provided you request it at least a few days in advance. Make sure you plan ahead before you travel, and stock up before you head to more remote areas. Due to animosity resulting from the Arab-Israeli conflict, many Muslim-majority countries do not have any official Jewish organizations, the main exceptions being Iran, Turkey, Bahrain and the Central Asia countries; you may want to stop in a neighboring country to stock up before you go. However, depending on how strict you are, halal food that follows the rule of not mixing milk and meat may suffice in a pinch. Some of these countries may also have underground Jewish organisations that operate under a "neutral", less politically sensitive name. For obvious reasons, they do not advertise their services openly, but you may be able to get in contact with them through your home country's Jewish organisation. The "Abraham Accords" between Israel and a number of Arab-Muslim countries have already led to some high-scale hotels announcing kosher options now that relations with Israel have been made official.
In Argentina, Buenos Aires has a large variety of kosher restaurants and supermarkets, including the only kosher McDonald's outside of Israel. Sao Paulo and Santiago also have a wide variety of kosher establishments, and a few other cities, such as Montevideo, Bogota, Lima, and Rio de Janeiro have at least one kosher establishment.
South Africa is generally the best place on the continent to find reliably kosher food. Besides South Africa, significant Jewish communities can also be found (perhaps surprisingly) in the North African Arab countries of Tunisia and Morocco.
Australia has a wide variety of kosher establishments in Sydney, Melbourne, Brisbane and Perth, but outside of those four cities, it gets more difficult to find food. The same applies in New Zealand, where, for instance, kosher meat is unavailable outside of one kosher shop in Auckland and another in Wellington. However, many products in supermarkets in both countries are indeed kosher but not labelled as such, and the local rabbinical authorities provide lists of what products one can consume.