Boy, that's a lot of topics for one very simple and interesting (yet complicated) question...
Number Theory encompasses several approaches; at its heart, Number Theory is the study of the properties of the natural numbers, but pretty soon one is led to consider other structures, such as the integers modulo $m$, the rational numbers, complex and real numbers, etc. One could even make a case that the field of complex analysis was born from arithmetic (number-theoretic) considerations.
One might roughly divide number theory into three large swathes: classic or elementary number theory; algebraic number theory; and analytic number theory. (One can also add a fourth: recreational number theory which, to quote Hendrik Lenstra, "is that branch of Number Theory that is too difficult for serious study.")
"Classic" or elementary number theory is the kind of stuff that Fermat was justly famous for: diophantine problems, divisibility questions, the kind of thing in the first section of Gauss's Disquisitiones Arithmeticae, with questions that refer almost exclusively to natural numbers, integers, or rational numbers. Usually, you stick to working with integers, rationals, and integers modulo $n$ for different $n$s. The basic kind of stuff.
Algebraic and analytic number theory were originally distinguished by the kinds of tools that were used in the study of arithmetic problems, but later also by the kinds of questions that were asked (questions that often arose because of the tools one was working on). Analytic number theory uses complex numbers to study arithmetic properties; the original proof of the Prime Number Theorem (a statement about the prime counting function) is a classic example. It uses analytical tools (limits, integrals, real and complex analysis, etc) to study and answers such questions.
Algebraic number theory, on the other hand, uses algebraic (rather than analytical) tools to study problems. A classic example is the study of Fermat's Christmas Theorem (a prime $p$ can be written as a sum of two squares if and only if $p=2$ or $p\equiv 1\pmod{4}$) using the Gaussian integers, $\mathbb{Z}[i]$, or the proofs of quadratic and cubic reciprocity that use the Gaussian and the Eisenstein integers; the study of Gauss sums, etc. A typical object of concern for algebraic number theory is the ring of integers of a finite field extension of $\mathbb{Q}$ (called a number field): start with a finite field extension of $\mathbb{Q}$, $K$, and consider all elements of $K$ that are roots of monic polynomials with integer coefficients. They form a fairly nice ring, with some very nice properties (they are the prototypical example of a Dedekind domain). You also study what one might argue is the algebraic counterpart of the real and complex numbers (relative to the rationals), the $p$-adic numbers $\mathbb{Q}_p$ (you can think of the reals as a way to "complete" the rationals with respect to the absolute value function; you can think of the $p$-adic numbers as a way to "complete" the rationals either with respect to a different kind of absolute value function, or alternatively as a way of making sense of an "infinite sequence of approximations" modulo higher and higher powers of $p$). Algebraic number theory now has its own "higher level offshoot", Arithmetic Geometry, which brings in tools of algebraic geometry to study number theoretic questions (think if "Algebra", "Geometry", "Analysis", "topology, "Number Theory", etc. as 'first-level subjects'; then you have algebraic number theory, algebraic topology, analytic geometry, etc., as 'second-level subjects.' Now we have algebraic arithmetic geometry, a 'third level subject').
In both algebraic and analytic number theory you very quickly are forced to consider irrational numbers, sometimes even transcendental numbers, simply because they are there. The subject of diophantine approximation is an example.
What properties of real numbers are studied in number theory? Well, to be glib, those that are "arithmetically relevant". It's pretty hard to single out certain properties as important and others as not important. Some come up, some do not.
Real numbers are studied from many viewpoints, not merely from number theory. In a sense, the ideas of real numbers (as the real line) are just as old as number theory (if you want to peg Euclid with the basics of number theory, or Diophantus).
As Zev notes, the real numbers (and the complex numbers) have such a rich structure that they are studied from all sorts of viewpoints (topology, order, logic, analysis, algebra, even discrete mathematics). Likewise for the complex numbers.
As to differences in how you study the reals and complex numbers in analysis, in number theory, and in algebra, well, the kinds of questions you are interested in are different. In number theory, it is seldom of any interest that the first order theory of the real numbers with addition and multiplication is decidable (a theorem of Taski's); but that's a pretty big deal in logic. Pretty much like how a study of the Middle Ages might differ depending on whether you are an economist, a historian (a historian interested in Europe; a historian interested in the Middle East; a historian interested in France; etc), a sociologist, an epidemiologist, etc. The different viewpoints all cover the same ground, and will often overlap (transmission of the black death was a key motor of drastic changes in the economies of Europe, giving you a connection between epidemiology and economy), but the viewpoints differ. Given how ubiquitous the real and complex numbers are, you'll encounter them all over the place, playing somewhat different roles in each and provoking somewhat different questions in each.