I have two passports/nationalities. How do I use them when I travel?

I have the same situation and have traveled extensively. There where a few reasons why to use two passports on one trip:

1. Exiting one country which I have a passport to and entering the other country to which I am a citizen (must use the right passport for each country).
2. Was running out of space for stamps on the passport (just lazy to get a new one)
3. Going out of one country where I had a visa and into another where I didn't need a visa on my other passport.

Usually I would try to use only one passport during a trip, if you must use two it's usually not a problem, you can just show both at check in (if needed) and at immigration (if asked).

I was pulled aside one time in Hong Kong, mostly since my first name is spelled differently in both passports, after a short explanation I was sent on my way.

It's only problematic if you hold a passport to a country that is refused entry (e.g. going to Malaysia with an Israeli passport) and try to enter with your other passport. I would highly advice against this.

Safe travels!

A wrinkle I just encountered is that some electronic passport readers appear to be linked to departure records. I was travelling from New Zealand to the UK, and have ePassports for both (case 1 of the accepted answer). I showed my UK passport to the airline agent checking me in, and then attempted to use my NZ passport in the passport reader at the immigration control point. It did not work, and I had to use my UK passport instead.

Most commonly this is an issue when travelling between the countries of your nationality, which has been well-answered by other people. Otherwise, it may be an issue if you have visa-free travel to a particular third country under one nationality and not the other - in which case, obviously, use the passport that you don't need a visa in.

There is the case where you're travelling to a country that one of your nationalities doesn't get on with, in which case you can travel on your other passport. The obvious examples here are North Korea on a South Korean passport and parts of the Arab world on an Israeli passport. I would strongly suggest not using a second nationality in cases where you're facing that level of hostility; if you're exposed then you could be in real physical danger or danger of arrest.

But there's another case where having multiple nationalities is useful.

On most Western passports, you can travel to Israel; you can also travel to most of the Arab world. But if you try entering certain Arab countries with an Israeli stamp in your passport, then you'll be refused entry.

If you have two passports, you can have stamps for Israel (and Jordan and Egypt if you enter across the land border from Israel) in one passport and those for the (rest of the) Arab world on the other. While some countries will issue you with two passports, it's a lot easier to keep them apart if you have two different nationalities (you're a lot less likely to pick up the wrong one by mistake).

The anser to your question can depend on the countries involved. For example, a US State Department website says this about dual citizenship Israeli-Americans:
"Israeli citizens naturalized in the United States retain their Israeli citizenship, and children born in the United States to Israeli parents usually acquire both U.S. and Israeli nationality at birth. Israeli citizens, including dual nationals, must enter and depart Israel on their Israeli passports."

Source: travel.state.gov