Let
$$\begin{align}
\overline{\mathbb{R}^{p,q}}~~:=&~~\left\{y\in \mathbb{R}^{p+1,q+1}\backslash\{0\}\mid \eta^{p+1,q+1}(y,y)=0\right\}/\mathbb{R}^{\times} \cr
~~\subseteq &~~ \mathbb{P}_{p+q+1}(\mathbb{R})~~\equiv~~(\mathbb{R}^{p+1,q+1}\backslash\{0\})/\mathbb{R}^{\times}, \cr
&\mathbb{R}^{\times}~~\equiv~~\mathbb{R}\backslash\{0\},
\end{align}\tag{1}$$
denote the conformal compactification of $\mathbb{R}^{p,q}$. Topologically,
$$ \overline{\mathbb{R}^{p,q}}~\cong~(\mathbb{S}^p\times \mathbb{S}^q)/\mathbb{Z}_2 .\tag{2} $$
The $\mathbb{Z}_2$-action in eq. (2) identifies points related via a simultaneous antipode-swap on the spatial and the temporal sphere.
The embedding $\imath: \mathbb{R}^{p,q}\hookrightarrow \overline{\mathbb{R}^{p,q}}$ is given by
$$\imath(x)~:=~\left(1-\eta^{p,q}(x,x): ~2x:~ 1+\eta^{p,q}(x,x)\right). \tag{3} $$
Let $n:=p+q$. [If $n=1$, then any transformation is automatically a conformal transformation, so let's assume $n\geq 2$. If $p=0$ or $q=0$ then the conformal compactification $\overline{\mathbb{R}^{p,q}}~\cong~\mathbb{S}^n$ is an $n$-sphere.]
On one hand, there is the (global) conformal group
$$ {\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}\tag{4}$$
consisting of the set globally defined conformal transformations on $\overline{\mathbb{R}^{p,q}}$. This is a $\frac{(n+1)(n+2)}{2}$ dimensional Lie group. The quotients in eqs. (2) & (4) are remnants from the projective space (1).
The connected component that contains the identity element is
$$ {\rm Conf}_0(p,q)~\cong~\left\{\begin{array}{ll} SO^+(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \} &\text{if both $p$ and $q$ are odd},\cr
SO^+(p\!+\!1,q\!+\!1) &\text{if $p$ or $q$ are even}.\end{array}\right.\tag{5}$$
The two cases in eq. (5) correspond to whether $-{\bf 1}\in SO^+(p\!+\!1,q\!+\!1)$ or not, respectively.
The global conformal group ${\rm Conf}(p,q)$ has 4 connected components if both $p$ and $q$ are odd, and 2 connected components if $p$ or $q$ are even.
The (global) conformal algebra $$ {\rm conf}(p,q)~\cong~so(p\!+\!1,q\!+\!1)\tag{6}$$
is the corresponding $\frac{(n+1)(n+2)}{2}$ dimensional Lie algebra. Dimension-wise, the Lie algebra breaks down into $n$ translations, $\frac{n(n-1)}{2}$ rotations, $1$ dilatation, and $n$ special conformal transformations.
On the other hand, there is the local conformal groupoid
$$ {\rm LocConf}(p,q)~=~\underbrace{{\rm LocConf}_+(p,q)}_{\text{orientation-preserving}} ~\cup~ \underbrace{{\rm LocConf}_-(p,q)}_{\text{orientation-reversing}} \tag{7}$$
consisting of locally defined conformal transformations. Let us denote the connected component that contains the identity element
$${\rm LocConf}_0(p,q)~\subseteq~{\rm LocConf}_+(p,q). \tag{8}$$
The local conformal algebroid
$$ {\rm locconf}(p,q)~=~{\rm LocConfKillVect}(\overline{\mathbb{R}^{p,q}})\tag{9}$$
consists of locally defined conformal Killing vector fields, i.e. generators of conformal transformations.
For $n\geq 3$, (the pseudo-Riemannian generalization of) Liouville's rigidity theorem states that all local conformal transformations can be extended to global conformal transformations, cf. e.g. this & this Phys.SE posts. Thus the local conformal transformations are only interesting for $n=2$.
For the 1+1D Minkowski plane we consider light-cone coordinates $x^{\pm}\in \mathbb{S}$, cf. e.g. this Phys.SE post. The locally defined orientation-preserving conformal transformations are products of 2 monotonically increasing (decreasing) diffeomorphisms on the circle $\mathbb{S}^1$
$$\begin{align}
{\rm LocConf}_+(1,1)~=~&
{\rm LocDiff}^+(\mathbb{S}^1)~\times~ {\rm LocDiff}^+(\mathbb{S}^1) \cr &~\cup~
{\rm LocDiff}^-(\mathbb{S}^1)~\times~ {\rm LocDiff}^-(\mathbb{S}^1),\cr
{\rm LocConf}_0(1,1)~=~&
{\rm LocDiff}^+(\mathbb{S}^1)~\times~ {\rm LocDiff}^+(\mathbb{S}^1).\end{align}\tag{10}$$
A orientation-reversing transformation is just a orientation-preserving transformation composed with the map $(x^+,x^-)\mapsto (x^-,x^+)$. The corresponding local conformal algebra
$$ {\rm locconf}(1,1)~=~{\rm Vect}(\mathbb{S}^1)\oplus {\rm Vect}(\mathbb{S}^1) \tag{11}$$
becomes two copies of the real Witt algebra, which is an infinite-dimensional Lie algebra.
For the 2D Euclidean plane $\mathbb{R}^{2}\cong \mathbb{C}$, when we identify $z=x+iy$ and $\bar{z}=x-iy$, then the locally defined orientation-preserving (orientation-reversing) conformal transformations are non-constant holomorphic (anti-holomorphic) maps on the Riemann sphere $\mathbb{S}^2=\mathbb{P}^1(\mathbb{C})$
$$\begin{align} {\rm LocConf}_0(2,0)~=~&{\rm LocConf}_+(2,0)\cr
~=~&{\rm LocHol}(\mathbb{S}^2), \cr\cr
{\rm LocConf}_-(2,0)~=~&\overline{{\rm LocHol}(\mathbb{S}^2)} ,\end{align}\tag{12}$$
respectively. An anti-holomorphic map is just a holomorphic map composed with complex conjugation $z\mapsto\bar{z}$. The corresponding local conformal algebroid
$$ {\rm locconf}(2,0)~=~{\rm LocHolVect}(\mathbb{S}^2)\tag{13}$$
consists of generators of locally defined holomorphic (sans anti-holomorphic!) maps on $\mathbb{S}^2$. It contains a complex Witt algebra.
References:
M. Schottenloher, Math Intro to CFT, Lecture Notes in Physics 759, 2008; Chapter 1 & 2.
R. Blumenhagen and E. Plauschinn, Intro to CFT, Lecture Notes in Physics 779, 2009; Section 2.1.
P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028; Chapter 1 & 2.
J. Slovak, Natural Operator on Conformal manifolds, Habilitation thesis 1993; p.46. A PS file is available here from the author's homepage. (Hat tip: Vit Tucek.)
A.N. Schellekens, CFT lecture notes, 2016.
