I am supposed to do a presentation on Lefschetz hyperplane section theorem via Morse theory (following Milnor's Morse Theory) for my algebraic geometry class...I more or less understand the proof, but I am really at a loss what could be good and easily presentable applications of the theorem. I am a beginner in algebraic geometry, so I can't do anything highly involved. Griffiths and Harris has a couple of examples....I was wondering if you people could suggest something really interesting and elegant... thanks a lot.

If you want more details than are in Milnor's book, I would recommend looking at AndreottiFrankel's paper (cited by Milnor as the source of his proof) or Nicolaescu's book on Morse theory. An important application is to the study of Stein manifolds. – Ian Agol Apr 1 '11 at 23:05
Dear anonymous, let $ X \subset \mathbb P^n(\mathbb C)$ be a smooth hypersurface of degree $d$ in projective space. Let me show how Lefschetz directly calculates the homology $H_i(X)=H_i(X,\mathbb Z)$ of $X$, except for $H_{n1}(X)$.
There is an embedding $v:\mathbb P^n(\mathbb C) \to \mathbb P^N(\mathbb C)$, called the Veronese embedding, mapping the original $P^n(\mathbb C)$ to a huge projective space $\mathbb P^N(\mathbb C)$ of dimension $N=\binom{n+d}{d} 1$
The image of $v$ is then an isomorphic copy $P$ of $\mathbb P^n(\mathbb C)$ and the charm of Veronese's embedding is that $v(X)$ is now a hyperplane section of $P $ i.e. $v(X)=P \cap H$ for some hyperplane $H \subset \mathbb P ^N(\mathbb C)$ .
You thus get $H_i(X) \simeq H_i(P)$ for $i\leq n2$ by applying Lefschetz for homology. For $i\geq n$ you deduce (by applying Poincaré duality and Lefschetz for cohomology ) $H_i(X) \simeq H^{2n2i}(X)\simeq H^{2n2i}(\mathbb P^n(\mathbb C))$. Since $P\simeq P^n(\mathbb C)$ all the above homology groups are easy to calculate. Indeed, for $0 \leq i \leq 2n$, projective space has homology $H_i(P^n(\mathbb C))$ alternately $0$ and $\mathbb Z$. Ditto for cohomology.
The only group which has partly escaped us is $H_{n1}(X)$, for which Lefschetz only tells us that it surjects onto $H_{n1}(P^n(\mathbb C))$. As a reality check, notice that for $n=2$ we know that the rank of $H_{1}(X)$ is $2g=(d1)(d2)$, where $g$ is the genus of our curve $X$ . This is indeed not equal to the rank of any (co)homology group of $P^2(\mathbb C)$ for $d \geq 3$
Optional complement: Allow me to show how concrete the Veronese embedding is. It maps the point
$(x_0:x_1:x_2:\cdots:x_n)\in \mathbb P^n$ to the point $(\cdots:M_\delta (x):\cdots) \in \mathbb P^N$ where $\delta = (d_0,\cdots,d_n) \quad d_0+\cdots d_n=d$ and $M_\delta (x)=x_0^{d_0}\cdots x_n^{d_n}$.
So that if for example $n=d=2$, the conic $x^2+2y^2+3y^24xy5xz6yz=0$ in $\mathbb P^2$ gets mapped isomorphically by Veronese onto the hyperplane section of the Veronese surface $v(\mathbb P^2)$ by the hyperplane $X+2Y+3Z4U5V6W=0$ of $\mathbb P^5$ . What could be simpler?
Here is an important application. It is probably in the relevant chapter of Voisins book.
Let $V^k \subset CP^n$ be a smooth projective variety, $d$ a number and $s$ be a section of the $d$th power of the Hopf bundle on $CP^n$. Assume that $s$ is transverse to the zero section and that $V$ and $s^{1}(0)$ are transverse. Let $W$ be the intersection. Then $\pi_i(W) \to \pi_i(V)$ is an iso if $i< k$. If $d$ were $1$, then this is just the hyperplane theorem. For larger $d$, the sections of $H^{\otimes d}$ define an embedding $f$ into a higherdimensional projective space $P^{N(d)}$ (the Veronese embedding, see Georges answer) and the section $s$ defines a hyperplane in that larger space. Then apply the hyperpalane theorem to $f(W) \to f(V)$; and you get the result.
A complete intersection variety $V^k \subset P^n$ is the common zero set of $nk$ sections $s_1, \ldots,s_{nk}$; $s_i$ a section in $H^{\otimes d_i}$ and these sections are in general position. By an iterative application you get that $\pi_i (V)=\pi_i(P^n)$ for $i< k$. Hence projective tori of dimension $>1$ are not complete intersections.