Number of Parameters of Lorentz Group We embed the rotation group, $SO(3)$ into the Lorentz group, $O(1,3)$ : $SO(3) \hookrightarrow O(1,3)$ and then determine the six generators of Lorentz group: $J_x, J_y, J_z, K_x, K_y, K_z$ from the rotation and boost matrices.
From the number of the generators we realize that $O(1,3)$ is a six parameter matrix Lie group.
But are there any other way to know the number of parameters of the Lorentz group in the first place?
 A: It's the same way you know there are three parameters in $SO(3)$. The equation $\Lambda^T \eta \, \Lambda = \eta$ has $(n^2+n)/2$ independent scalar equations. To see this, write the equation in component form: $\Lambda^{\mu\nu} \Lambda_\mu{}^\rho = \eta^{\nu\rho}$. Now we see there are $n^2$ scalar equations equations, but because $\eta$ is symmetric and the left hand side is symmetric in $\nu$ and $\rho$ as well, the equations related by switching $\nu$ and $\rho$ are the same. Thus we have established that there are  $(n^2+n)/2$ independent scalar equations. 
Since $\Lambda$ has $n^2$ components, we get $n^2-(n^2+n)/2 = n(n-1)/2$ degrees of freedom. In $3D$ this comes out to $3\cdot 2 /2=3$, and in $4D$ this comes out to $4 \cdot 3/2=6$.
A: You've got two very good answers from Hunter and NowIGetToLearnWhatAHeadIs.  However, it's probably useful to know that this beast $O(1,3)$ is isomorphic or locally isomorphic (i.e. has the same Lie algebra) to a surprising number of other interesting groups, which each give you a slightly different way to think about it. First note that its identity connected component $SO^+(1,3)$ of orthochronous, proper Lorentz transformations (those that keep the orientation of space and time the same, also called the "restricted" Lorentz group) of course determines the Lie algebra. 


*

*$SO^+(1,3)\cong {\rm Aut}(\hat{\mathbb{C}}) \cong PSL(2,\mathbb{C})$ is isomorphic to the Möbius group of all Möbius transformations, in turn isomorphic to the group of all conformal transformations of the unit sphere. So it is defined by $z\mapsto \frac{a\,z+b}{c\,z+d}$ with $a,\,b,\,c,\,d\in\mathbb{C}$ and $a\,d-b\,c=1$. So there are three independent complex parameters, i.e. six independent real parameters;

*The double cover of $PSL(2,\mathbb{C})$, namely $SL(2,\mathbb{C})$ (still locally isomorphic to $SO^+(1,3)$) is the group of all $2\times 2$ matrices of the form:
$$\exp\left(\frac{1}{2}\left[\left(\eta^1 + i\theta \gamma^1\right) \sigma_1 + \left(\eta^2 + i\theta \gamma^2\right) \sigma_2 + \left(\eta^3 + i\theta \gamma^3\right) \sigma_3\right]\right)$$
where $\sigma_j$ are the Pauli spin matrices, $\theta$ is the angle of rotation, $\gamma^1,\,\gamma^2,\,\gamma^3$ are the direction cosines of the rotation axis and $\eta^1,\,\eta^2,\,\eta^3$ the components of the rapidities of the Lorentz transformation. So it's just like the general matrix $\exp\left(\frac{\theta}{2}\left(\gamma^1 \sigma_1 + \gamma^2 \sigma_2 + \gamma^3 \sigma_3\right)\right)$ in $SU(2)$ but with three complex parameters, rather than three real ones ($\theta \gamma^j$) for $SU(2)$. So again we see six real parameters.
A: From special relativity we know that a Lorentz transformation:
\begin{equation}
x'^\mu = \Lambda^\mu {}_\nu x^\nu
\end{equation}
preserves the distance:
\begin{equation}
 g^{\mu \nu} \Delta x_\mu \Delta x_\nu = g^{\mu \nu} \Delta x_\mu' \Delta x_\nu'
\end{equation}
The above two equations imply:
\begin{equation}
 g^{\mu \nu} = g^{\rho \sigma}\Lambda_\rho {}^\mu \Lambda_\sigma {}^\nu
\end{equation}
Now, let us consider an infinitesimal transformation:
\begin{equation}
\Lambda_\nu {}^\mu = \delta_\nu{}^\mu + \omega_\nu{}^\mu + O(\omega^2)
\end{equation}
such that we can write:
\begin{equation}
\begin{aligned}
g^{\mu \nu} & = g^{\rho \sigma}\Lambda_\rho {}^\mu \Lambda_\sigma {}^\nu \\&
=  g^{\rho \sigma} \left( \delta_\rho{}^\mu + \omega_\rho{}^\mu + \cdots \right)\left( \delta_\sigma{}^\nu + \omega_\sigma{}^\nu + \cdots \right) \\&
= g^{\mu \nu} + g^{\mu \sigma} \omega_\sigma{}^\nu + g^{\rho \nu} \omega_\rho{}^\mu + O(\omega^2) \\&
=  g^{\mu \nu} + \omega^{\mu\nu} + \omega^{\nu \mu} + O(\omega^2)
\end{aligned}
\end{equation}
and so:
\begin{equation}
\omega^{\mu\nu} = - \omega^{\nu \mu}
\end{equation}
Thus, the matrix $\omega$ is a $4 \times 4$ antisymmetric matrix, which corresponds to $6$ independent parameters (i.e. the $3$ parameters corresponding to boosts and the $3$ parameters corresponding to rotations).
