In this context, “international” means you are a graduate student or a postdoc in a country in which you are not a citizen or legal permanent resident. People in this position have some special tasks and concerns that citizens do not and that will have a large impact on their job searches and their finances. If you don’t see the answer to your question here, you may email it to me at firstname.lastname@example.org or find me on twitter @akrook.
The first thing you have to do at all times is to understand the terms of your visa or other residency permit and to keep any required documentation current. You should not assume that you will be able to change those terms, say, to extend a visa allowing you to study for a Ph.D. to cover an additional two years of postdoctoral work or other employment. If you believe you may want to do that, you need to start that process with your institution and the relevant agencies of the country you are visiting and the one where you make your permanent home. You will also need a backup plan in case the answer to your request for an extension is “no.” (By the way, international students, faced with hard deadlines, tend to finish their Ph.D. programs more quickly than citizens.)
Especially if you believe you may return to your home country when you are done with your work, stay in touch with friends and colleagues there, and take time to make connections in your field there. It is harder to start or conduct a job search remotely, and it will require more work from you to stay connected and to create a professional network than it will for someone who lives there and can do so casually. For that same reason, it’s worth seeing if there is a group of students from your home country or region at your institution. You will need connections as you start searching for a job, and that community may be helpful to you. Remember that remote job searches take longer than local ones, and that to get offers you will probably have to travel for final interviews, with financial and time implications for your work.
Your status as visitor may complicate your financial position at your institution. Aside from the increased costs international students tend to face (travel expense to visit family, clothing suited to the climate you are visiting if different from your own, etc.) some funds available to citizens may not be available to you. For example, if your institution assumes that, after your fellowship year(s) and teaching year(s) you will apply for a competitive dissertation grant to cover at least one year’s work, check to see that those grants are open to non-citizens. You may be restricted to a subset of them, and if all the international students apply to that same pool, you chances of winning one may be lower than others’. You may also be less able to supplement your stipend with casual employment, especially non-academic casual employment, which your visa may not permit.
Your job search will be different than your peers’. If you are seeking academic work in your home country, make sure you know all the expectations for your application and faculty members’ support. Find how, for example, how many references you need, in what language(s) they may be written, whether your thesis requires formal posting and translation. If you are working in an English-language program, especially in the humanities, don’t assume that the ubiquity of English means that it will be accepted in your application without translation. It may be accepted, but check. If you are seeking academic work in the country where you are studying or where you are a noncitizen, check job postings for your eligibility, especially in grant-dependent fields where governments may specify who can be hired on to the grant-funded projects. And if you get such a job, be prepared for some resentment from fellow students who are citizens but don’t get jobs. It may not come, but be prepared for it.
If you are seeking nonacademic work in your home country or the country where you study, find out the top twenty or so companies (or nonprofits, or NGOs, depending on what you want to do) from your home country that work in your academic host country, and the same number from your academic host country that work in your home country. Those entities are likeliest to value your experiences in both countries and are good starting places to look for work. You may also find that your consulate or embassy has some information on entities in both countries looking for good employees.