$X$ is defind to be the functor $R$-alg$\to Sets$ where for any $R$ algebra $C$, $X(C)$ is the set of formal power series $f(t) =\sum_{n≥0} a_nt^n ∈ \Bbb C[[t]]$ such that $f(t)^2 = 1 + t$.
I think it is not an affine scheme because if $X\cong Spec(A)$ for some $R$-algebra $A$, we require $A=R[x_s|s\in S]/\langle T\rangle$ where $T$ is the set of the relations of form $\{\sum_{n=0}^\infty(a_nt^n)^2-1-t\}$, but these "relations" are "infinite polynomials" so it is not in the polynomial ring $R[t]$. But I still do not know whether even this form of relation is plausible to be used to define an affine scheme. Could someone please tell me whether this functor is an affine scheme or not? Thanks in advance!
Your "infinite" relation is actually just an infinite conjunction of ordinary finite relations. Namely, the equation $\left(\sum a_nt^n\right)^2-1-t=0$ just means that each coefficient of this formal power series is $0$: $a_0^2-1=0$, $2a_0a_1-1=0$, $a_1^2+2a_0a_2=0$, and so on. So your functor $X$ is an affine scheme, and is represented by the $R$-algebra $R[a_0,a_1,\dots]/(a_0^2-1,2a_0a_1-1,a_1^2-2a_0a_2,\dots)$ where there are infinitely many relations which say that all of the coefficients of the power series $\left(\sum a_nt^n\right)^2-1-t$ are $0$.