A function $f$ is said to be "even" if and only if for every $x$ in the domain, $f(x)=f(-x)$. The simplest way to check this is to verify equality after you plug in. To verify that a function is not "even" we need to exhibit a particular number $x$ for which $f(x)$ and $f(-x)$ are distinct.
For instance, with $f(x)=\tan(x)$, $f(x)$ is not even: if $x=\frac{\pi}{4}$, then $f(x) = f\left(\frac{\pi}{4}\right) = 1$ but $f(-x) = f\left(-\frac{\pi}{4}\right) = -1 \neq f(x).$
On the other hand, $f(x) = \sqrt{a^2-x^2}$ is even: the domain is $[-|a|,|a|]$ (since we are assuming $a=1$, this would be $[-1,1]$. If we plug in $-x$ instead of $x$, we have: $f(-x) = \sqrt{a^2 - (-x)^2} = \sqrt{a^2 - x^2} = f(x),$ and this holds for all $x$, so $f$ is even.
Check the other functions; (by the way, if you have values of $x$ that are in the domain but $-x$ is not in the domain, then the function is not even.)
"Odd" is similar, except that the definition is that for all $x$ in the domain, we must have $f(-x) = -f(x)$.
For example, $f(x)=\tan(x)$ is odd, since for all $x$ in the domain, we have $f(-x) = \tan(-x) = \frac{\sin(-x)}{\cos(-x)} = \frac{-\sin(x)}{\cos(x)} = -\frac{\sin(x)}{\cos(x)} = -\tan(x) = -f(x).$ (Using the fact that $\sin(x)$ is odd and $\cos(x)$ is even).
A function $f$ is periodic if there exists $p\gt 0$ such that $f(x)=f(x+p)$ for all $x$. For example, $\sin(x)$ is periodic, since $\sin(x+2\pi) = \sin(x)$ for all $x$. On the other hand, $f(x) = 2^x$ is not periodic, since we know that for all real numbers $a$ and $b$, if $a\lt b$ then $2^a\lt 2^b$, so we can never "repeat values". The number $p$ would be a period of the function. (Generally, there are many periods, since if $p$ is a period, then so is $2p$, and $3p$, and $4p$, etc.
A function $f(x)$ is one to one if $a\neq b$ implies $f(a)\neq f(b)$ (equivalently, if whenever $f(a)=f(b)$, then it must be the case that $a=b$). You can prove that a function is not one-to-one by exhibiting a single pair of numbers $a\neq b$ where the function takes the same value. For example, $f(x)=x^2$ is not one-to-one, because even though $1\neq -1$, we nevertheless have $f(1)=f(-1)$. On the other hand, $f(x) = x^3$ is one-to-one, because if $f(a)=f(b)$, then that means that $a^3=b^3$, and the only way this can occur is if $a=b$.
All of this is algebraically.
Geometrically, assuming you can get nice and accurate graphs, a function $f(x)$ is even if the graph of $y=f(x)$ is symmetric about the $y$ axis; the function is odd if it is symmetric about the origin. It is periodic if it "repeats" after a finite length (think about the graph of $y=\sin(x)$). And it is one-to-one if it passes the "horizontal line test":
Horizontal Line Test The graph of $y=f(x)$ intersects each horizontal line in at most one point.