Algebra and Analysis
Algebra is about combining things together with operations while analysis focuses more on studying the closeness or "connectedness" between points.
Some of your confusion might stem from the fact that algebra and analysis can often work together. If you take courses in abstract algebra and real analysis, you will study each topic on its own, mostly divorced from the other.
To expand on algebra a little...
High school algebra involves addition and multiplication of real numbers and the properties of these operations. For example, if $ab=0$, then you know either $a=0$, $b=0$, or both.
In later algebra courses you would learn about more interesting algebraic structures that have different rules. If you have studied linear algebra you would know that, unlike real number multiplication, matrix multiplication doesn't commute in general, $AB \neq BA$, and knowing $AB=0$ doesn't imply that $A$ or $B$ is the zero matrix. You could also study algebraic rules for complex numbers, polynomials, and many other structures. (Quaternions are another example where multiplication doesn't commute.)
In an abstract algebra class you will forget about "what it is" you are studying and focus your study on the operations used to combine objects. In a way you are studying how things behave rather than what they are. This leads to the idea of groups, rings and fields among other structures which let you recognize common algebraic patterns throughout all areas of math instead of focusing on concrete specifics.
I have less experience in analysis, but I can say a few things...
You would already be familiar with some of the ideas of analysis from calculus. Many calculus classes don't give a rigorous definition of a limit and simply assume it works based on intuition (indeed, this is how calculus was developed historically).
A study of analysis would revisit limits and give a rigorous definition what it means for a series to "converge" and what it means for a function to be "continuous". From here one can develop rigorous definitions for derivatives and integrals.
Later these ideas will be generalized to work in $n$-dimension space, for functions of complex variables, and any spaces in which the idea of "continuity" of functions makes sense. (These spaces go by the name of topological spaces).
After studying them separately, analysis and algebra tend to show up together all over the place. A simple example might be the product rule from calculus, which is an algebraic rule that results from studying analytical properties of functions:
$$d(fg) = df g + fdg$$
Another example is the inner product, which grew out of the dot product from linear algebra but can also be applied to pairs of continuous functions.
Abstract algebra is largely (but not only) about sets with operations and their properties.
Mathematical analysis is largely (but not only) more about topology, measure, and how you can apply topology and measure to functions, namely integration and differentiation.
Contrary to your comment about continuity and discreteness, there are many interesting algebraic structures with topologies that you can study. For example, there are rings of continuous functions from a topological space into itself.
Similarly, analysis is no stranger to discreteness. For example, summable sequences are nothing more than integrable functions from $\Bbb N\to \Bbb R$ where the measure is the counting measure.
Functional analysis (at the very least) is a complex marriage of the two disciplines. There is no reason to draw a line between the two disciplines as you described.
You need to know both. Honestly at higher levels of math things aren't broken down into analysis or algebra. There's functional analysis. Algebraic topology etc.
Both are important to know.
The real question however, is how do you eat your corn?