While lithium is generally deficient in the atmosphere of evolved giant stars because of the efficient mixing-induced dilution, a small fraction of red giants show unusually strong Li lines indicative of conspicuous abundance excess. With an aim to shed light on the origin of these peculiar stars, we carried out a spectroscopic study on the observational characteristics of selected 20 bright giants already known to be Li-rich from past studies, in comparison with the reference sample of a large number of normal late G -early K giants. Our special attention was paid to clarifying any difference between the two samples from a comprehensive point of view (i.e., with respect to stellar parameters, rotation, activity, kinematic properties, 6 Li/ 7 Li ratio, and the abundances of Li, Be, C, O, Na, S, and Zn). Our sample stars are roughly divided into "bump/clump group" and "luminous group" according to the positions on the HR diagram. Regarding the former group (1.5 < ∼ log(L/L ⊙ ) < ∼ 2 and M ∼1.5-3 M ⊙ ), Li-enriched giants and normal giants appear practically similar in almost all respect except for Li, suggesting that surface Li enhancement in this group may be a transient episode which normal giants undergo at certain evolutionary stages in their lifetime. Meanwhile, those Li-rich giants belonging to the latter group (log(L/L ⊙ ) ∼ 3 and M ∼3-5 M ⊙ ) appear more anomalous in the sense that they tend to show higher rotation as well as higher activity, and that their elemental abundances (especially those derived from high-excitation lines) are apt to show apparent overabundances, though this might be due to a spurious effect reflecting the difficulty of abundance derivation in stars of higher rotation and activity. Our analysis confirmed considerable Be deficiency as well as absence of 6 Li as the general characteristics of Li-rich giants under study, which implies that engulfment of planets is rather unlikely for the origin of Li-enrichment.