A small radio frequency band used in Europe to provide added network capacity for GSM 900 networks.
If you are using one of the European GSM carriers that uses EGSM, then using an EGSM phone may improve coverage and/or reduce "network busy" messages.
EGSM spectrum is 880-890 MHz paired with 925-935 MHz, which is just below the original GSM 900 band.
Phones described as having "EGSM" or "EGSM 900" support both the orginal GSM 900 band and the extended band. Older phones with "GSM 900" may not support EGSM. Most newer phones with "GSM 900" do support EGSM, it is just not listed that way since it is assumed that newer phones support it.