If you go outside at night during spring or fall migration you may hear very short tseeps and chirps coming from overhead. These are the calls of migrating birds. Most of the calls by far will be made by warblers, sparrows and thrushes, but you may also hear other birds such as Rose-breasted Grosbeak, Scarlet Tanager, or even cuckoos, shorebirds, herons or rails. Some birds, such as flycatchers, also migrate at night, but rarely, if ever, call.
This page gives a little guidance on how to observe flight calls. You can just go outside and listen, of course, or you can use a microphone to listen from indoor comfort, or you can record them.
Identifying flight calls
The best source for learning the flight calls of birds in eastern North America is the multimedia CD-ROM Flight Calls of Migratory Birds by William R. Evans and Michael O'Brien, which you can order from oldbird.org. The CD-ROM has both recordings and sonograms of the flight calls.
Typical sonograms of some of the most commonly heard flight calls are available at http://oldbird.org/Library.htm. However, the calls do vary, and the CD-ROM has many more examples of calls by each species.
If you know of similar resources for other regions of the world, please send me a note about them.
When migrating birds call
These two graphs show the times of night when high-frequency and mid-frequency calls were recorded. The mid-frequency calls are plotted as hours before sunrise to make clear the increase in thrush calls in the hour or so before dawn. The high frequency calls are characteristic of warblers and sparrows and the mid-frequency calls are made by thrushes and other mid-size birds

Microphones:
For an introduction to microphones and recording avian flight calls go to oldbird.org.
The Oldbird "flower pot" mic works well, is easy to build, and is inexpensive. I don't favor using a parabolic mic for recording nocturnal flight calls, because they have a narrow focus and will tend to miss calls coming from the side.
If you just intend only to listen you might need a preamp between the mic and your stereo or other listening device, unless the device has a built-in preamp as audio recorders do.

A migration microphone needs to be placed where its "view" of the sky is unimpeded and interfering noises, which are primarily made by insects, amphibians and humans, are minimized. A roof top away from vegetation is a good possibility.
Recording:
There are two basic options: you can record directly to your computer or your can use a stand alone recorder and transfer the files to your computer for analysis. I am not going to suggest specific models of equipment, because the market constantly changes, and what I say today will be obsolete tomorrow.
No matter which method you choose, you need to be able to produce 16 bit monophonic files at 22050 samples per second. (This is sometimes referred to as A/D resolution = 16 bit/22.05kHz.) There are three good reasons for this. First, you don't need better resolution for this purpose. Second, the free software for detecting and extracting the individual calls into separate sound files requires this format, and you will have to convert your files in order to use these programs if you use a different file format. Third, the sound files will be very large, and it is simply easier to store, transfer and edit smaller files. The file from an overnight, monophonic, 16 bit/22.05kHz recording will be > 1GB. A stereo file would be twice as large and a 44.1kHz file would be twice as large again.
Stand alone recorders: There many relatively inexpensive, good quality recorders these days. Get one that is able to record 16 bit monophonic files at 22050 samples per second.
Recording to your hard drive: There are two ways to do this. The first is to connect your microphone to a preamplifier which is connected to the input jack of your sound card. You use your recording software to set the A/D resolution to 16 bit/22.05kHz. The second is to use a USB preamp. In this case, the microphone is connected to the USB preamp which is connected to a USB port on your computer. The A/D converter is in the USB preamp box, so you need to be sure that it is capable of 16 bit/22.05kHz A/D resolution. Some people think that the USB preamp should provide a better recording, because the A/D conversion takes place outside the electrically noisy computer. I don't know if this is true.
I'd avoid analog recording systems such as VCR tape. The problem isn't quality but awkwardness, because playing back the tape is slow.
Software:
Audacity is a good, all-purpose audio editing program that produces sonograms and also will record microphone input to your hard drive. It is free.
Raven Lite is another free audio program that produces sonograms and will record input to your hard drive.
Eventually you may want to use your computer to extract flight calls from your recordings and save them as individual sound files in order to sort them by species. Tseep-X and Thrush-x work well for extracting the calls (I think that Tseep-X is more efficient than Thrush-X - i.e. Tseep-x extracts a higher percentage of its target calls than Thrush-x - but I haven't finished studying this question.) Tseep-R and Thrush-R are similar to Tseep-X and Thrush-x, but they extract the calls 'on the fly' from the sound stream coming in and the rest of the data is discarded. Thus, you can only collect one class of calls in a session if you use Tseep-R or Thrush-R.
Once you have extracted your calls as individual files, Glassofire lets you look at many sonograms at once, listen to them and sort them into different classes such as noise and various bird species. It's extremely useful.