SSB: Single SideBand (J3E) is a refinement of amplitude modulation (AM) that more efficiently uses electrical power and bandwidth by suppression of either the upper sideband or the lower sideband as well as suppression of the carrier, resulting in only a singe sideband (SSB). Amplitude modulation (AM) produces a modulated output signal that has twice the bandwidth of the original baseband signal. Therefore, single-sideband modulation avoids this bandwidth doubling, and the power wasted on a carrier. Typical SSB audio bandwidth is less than 3kHz in audio frequency response and is designated by the FCC as J3E mode.

eSSB: Extended Single SideBand is any J3E SSB transmission that exceeds the audio bandwidth of standard or traditional 2.9kHz SSB J3E modes (ITU 2K90J3E) starting at 3kHz (ITU 3K00J3E), in order to support the fidelity required and desired for relative high fidelity, full range clean and articulate vocal audio. eSSB is also a refinement of amplitude modulation (AM) that more efficiently uses electrical power and bandwidth as opposed to AM, but can still exhibit the same audio frequency response of broadcast AM if desired.
(Source: http://www.nu9n.com/essb.html)

As the eSSB definition above states, it is an "Extended" version of SSB. The radio frequency and audio frequency footprint (bandwidth) is extended as a result of several factors; I.F. filter modifications, and/or Digital Signal Processing, and external audio pre-emphasis processing via Equalizing, Compressing, Peak-Limiting, etc. of the source audio input. (See the tables below:)