Mirza,

You had several bugs/typos:

1. Y and y are different variables -- be careful of capitalization
2. you need a list of equations -- not a list of lists
3. you use NeumannValue incorrectly -- it is not used in the boundary conditions it is in the equation itself (even though it acts at the boundary)
4. you use extra lists in places they do not belong (i.e inside of DirichletCondition, the integration variable, etc.)

this computes a solution:

eq1 = {D[Subscript[\[Theta], c][x, y], x] - 
     2/(y - y^2)*D[Subscript[\[Theta], c][x, y], y, y] == 
    NeumannValue[0, y == 1]};
ic = {Subscript[\[Theta], c][0, y] == 1};
bc = {DirichletCondition[Subscript[\[Theta], c][x, y] == 0, y == 0]};

sol = NDSolve[Join[eq1, ic, bc], 
  Subscript[\[Theta], c], {x, 0, 1}, {y, 0, 1}]

Regards,

Neil

When I copy eq1 into Mathematica, I get a message

Syntax::bktmcp: Expression "{D[Subscript[\[Theta],c][x,y],x]-2/(Y-Y^2)*D[Subscript[\[Theta],c][x,y],y,y]==0" has no closing "}".