You must add a list of variables in Solve[] function, something like this:

eq1 = (IDSAT - ((1/2*mun0*Coxn*W/L)*((VGS - Vth)^2)*(1 + 
         lambdan*VDS)) == 0);
eq2 = (IRSS - (VX/RSS) == 0);
eq3 = (IRD - (VDD/RD) + (VOUT/RD) == 0);
eq4 = (VGS - VG + VX == 0);
eq5 = (VDS - VOUT + VX == 0);
eq6 = (VG - VIN == 0);
eq7 = (IDSAT - IRD == 0);
eq8 = (IRD - IRSS == 0);

eqns = {eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8};
Solve[eqns, {VOUT, IRD, IRSS, VX, VGS, VDS, VG, IDSAT}]

or:

eq1 = (ID == ((0.5mun0CoxnW/L)((VGS - Vth)^2)(1 + lambdanVOUT))); eq2 = (IRD == ID); eq3 = (IRD == (VDD - VOUT)/RD);

eqnsp2 = {eq1, eq2, eq3}; Solve[eqnsp2, {VOUT, IRD, ID}]