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We report the discovery of a peculiar X-ray transient, EP240408a, by Einstein Probe (EP) and follow-up studies made with EP, Swift, NICER, GROND, ATCA and other ground-based multiwavelength telescopes. The new transient was first detected with Wide-field X-ray Telescope (WXT) on board EP on April 8th, 2024, manifested in an intense yet brief X-ray flare lasting for 12 s. The flare reached a peak flux of 3.9 × 10−9 erg cm−2 s−1 in 0.5–4 keV, ∼300 times brighter than the underlying X-ray emission detected throughout the observation. Rapid and more precise follow-up observations by EP/FXT, Swift and NICER confirmed the finding of this new transient. Its X-ray spectrum is non-thermal in 0.5–10 keV, with apower-law photon index varying within 1.8–2.5. The X-ray light curve shows a plateau lasting for ∼4 d, followed by a steep decay till becoming undetectable ∼10 d after the initial detection. Based on its temporal property and constraints from previous EP observations, an unusual timescale in the range of 7–23 d is found for EP240408a, which is intermediate between the commonly found fast and long-term transients. No counterparts have been found in optical and near-infrared, with the earliest observation at 17 h after the initial X-ray detection, suggestive of intrinsically weak emission in these bands. We demonstrate that the remarkable properties of EP240408a are inconsistent with any of the transient types known so far, by comparison with, in particular, jetted tidal disruption events, gamma-ray bursts, X-ray binaries and fast blue optical transients. The nature of EP240408a thus remains an enigma. We suggest that EP240408a may represent a new type of transients with intermediate timescales of the order of ∼10 d. The detection and follow-ups of more of such objects are essential for revealing their origin. [ABSTRACT FROM AUTHOR]