Giving variables in a coordinate ring different weights I am reading a book and I am curious about a certain notion. 
Consider $R = k[x_1,x_2,x_3,x_4,t]$ and let $G = \{\underbrace{x_1 x_3-x_2^2 + t x_3^2}_{f_1}, \underbrace{x_1 x_4-x_2 x_3 +t x_2^2}_{f_2} \}$. 
When $t=0$, we obtain the standard twisted cubic. 
After imposing the following weights to the variables: 
$$  
wt(x_i)=1 \mbox{ and } wt(t)=0. 
$$
does this mean I am viewing $t$ as a constant, and would you say $f_1$ and $f_2$ are homogeneous of degree 2, rather than think of it as a mixed degree polynomial? 
$$ $$
What if I, instead, impose the weights to be 
$$ 
wt(x_i)=2 \mbox{ and } wt(t)=1? 
$$ 
What is the purpose of giving variables different weights? 
$$ 
$$ 
Addendum: is there a geometric significance to the notion of weights?
 A: Setting different weights to a variable is changing the fact wether a polynomial is homogeneous or not. For a grading you might want to fix a factor $G$ of $\mathbb{Z}^n$ and give your indeterminates weights $g \in G$. After that you can talk about homogeneous elements and wether the ring or a module over it is generated in certain degree. You can also talk about wether a ring homomorphism is one of graded rings or not.
For me the main thing that changes is the set of homogeneous prime ideals of a graded ring $S$, the so called ${\rm proj}(S)$.
If K is an algebraically closed field and you have a look at the ring
$$
S := K[x_0,x_1,x_2,x_3]
$$
and take $G =\mathbb{Z}$ and ${\rm deg}(x_i) = 1$ for all $i$, then you get
$$
{\rm proj}(S) \cong \mathbb{P}^2.
$$
If you take $G = \mathbb{Z}^2$ and set ${\rm deg}(x_0)={\rm deg}(x_1)=(1,0)$ and ${\rm deg}(x_2)={\rm deg}(x_3)=(0,1)$ you get
$$
{\rm proj}(S) \cong \mathbb{P}^1 \times \mathbb{P}^1 \not\cong \mathbb{P}^2.
$$
A: Yes. No. It is natural to do in certain arguments (e.g. to make certain polynomials one cares about homogeneous). Geometrically weights correspond to actions of the multiplicative group $k^{\ast}$ (for sufficiently large $k$); that is, giving $x_i$ a weight of $e_i$ corresponds to letting $a \in k^{\ast}$ act by $x_i \mapsto a^{e_i} x_i$. 
For the general notion, see graded algebra. 
