What are called the standard conjectures on algebraic cycles are several conjectures brought up by Grothendieck, concerned with the relation between algebraic cycles and Weil cohomology theories. The first (Lefschetz standard conjecture) is an existence assertion for algebraic cycles (Grothendieck - 68), the second (Hodge standard conjecture) is a statement of positivity, generalising Weil’s well-known positivity theorem in the theory of abelian varieties (Grothendieck - 68). They were also followed by “Beilinson's dream” on motivic (complexes of) sheaves which comprise what are called the standard conjectures of Beilinson.
A. Grothendieck, Standard Conjectures on Algebraic Cycles, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford University Press, pp. 193–199, pdf
Steven Kleiman, Algebraic cycles and the Weil conjectures, in: Dix exposés sur la cohomologie des schémas, North-Holland, pp. 359–386, 1968, MR0292838; The standard conjectures, Motives (Seattle, WA, 1991), Proc. of Symposia in Pure Math. 55, American Mathematical Society, pp. 3–20, 1994, MR1265519