Explicit eigenvalues of the Laplacian
Besse (1978, p.202) has the spectra of compact rank 1 symmetric spaces (CROSSes). In addition to $\mathrm S^n$ due apparently to Heine (1863, §19; 1878, §128), this gives $\mathbf{RP}^n$, $\mathbf{CP}^n$, $\mathbf{HP}^n$ and $\mathbf{OP}^2$.
Edit: Also, for $M$ a compact semisimple Lie group it is well known (due apparently to Freudenthal (1954))1 that the Laplacian (= Casimir) acts on the $\lambda$-subspace in the Peter-Weyl decomposition $L^2(M)=\smash{\bigoplus_\lambda V_\lambda^{\phantom*}\!\otimes V_\lambda^*}$ by the scalar $c(\lambda):=\smash{\|\lambda + \rho\|^2-\|\rho\|^2}$; so these are its eigenvalues. $(\lambda$: dominant weight; $\rho=\frac12\!\sum\limits_{\alpha > 0}\alpha$; $\|\cdot\|$: Killing norm.)
Further edit: The literature contains quite a few more cases than the answers so far. As no single source or search word easily returns them, I list here what I found (others’ answers not repeated):
First, the Casimir method above extends to give the spectrum of the normal metric on $G/H$ ($G$ compact semisimple, $H$ closed). In fact, by Frobenius reciprocity, $V_\lambda$ occurs in $L^2(G/H) = \operatorname{Ind}_H^G1$ with multiplicity equal to the dimension of $V_\lambda{}^H=\{H$-fixed vectors in $V_\lambda\}$. So the eigenvalues are exactly all $c(\lambda)$ for $\lambda$ such that $V_\lambda{}^H\ne0$. After spheres, this method was applied to:
Stiefel manifolds $\mathrm{SO}_n\,/\,\mathrm{SO}_{n-m}$ by Levine (1969, p.519), Gelbart (1974), Strichartz (1975).
CROSSes ($\mathbf{RP}^n$, $\mathbf{CP}^n$, $\mathbf{HP}^n$, $\mathbf{OP}^2$) by Berger & al. (1971, pp.159-173), Cahn & Wolf (1976).
Flag manifolds $G\,/\,T$ ($T$: maximal torus) by Yamaguchi (1979, p.110).
Grassmannians $\mathrm{Gr}_2(\mathbf R^n)$ by Strese (1980, p.78) and Tsukamoto (1981).
Aloff-Wallach spaces $\mathrm{SU}_3\,/\,\mathrm S^1$ by Urakawa (1984, p.984) and Joe et al. (2001, p.417).
Symmetric spaces $\mathrm{SU}_n\,/\,\mathrm{SO}_n$ by Gurarie (1992, p.253).
Grassmannians $\mathrm{Gr}_n(\mathbf C^{n+m})$ and $\mathrm{SU}_{n+m}\,/\,\mathrm{SU}_n\times\mathrm{SU}_m$ by Ben Halima (2007, pp.546, 549).
Secondly, some cases yield to other methods:
Lens spaces $\mathrm S^{2n-1}\,/\,\mathbf Z_p$ by Sakai (1976, p.256).
Hopf manifolds $M_\alpha$ by Bedford & Suwa (1976, p.261).
Berger spheres (total spaces of the Hopf fibration $\mathrm S^1\to\mathrm S^{2n+1}\to \mathbf{CP}^n$ with rescaled fiber) by Tanno (1979, p.184).
Jensen spheres (total spaces of the Hopf fibration $\mathrm S^3\to\mathrm S^{4n+3}\to \mathbf{HP}^n$ with rescaled fiber) by Tanno (1980, p.103) and Nilsson & Pope (1983, p.68).
Grassmannians $\mathrm{Gr}_2(\mathbf C^n)$ by Sumitomo & Tandai (1985, p.153).
Riemannian two-step nilmanifolds $G\,/\,\Gamma$ by Pesce (1993).
1 Note added: Rogawski–Varadarajan (2012, p. 690) attribute the formula for $c(\lambda)$ to Casimir–van der Waerden (1935; note the reviewer). However, I’m not sure I can find it there...?
You can compute the eigenvalues explicitly for flat tori $\mathbb{R}^n/\Gamma$, where $\mathbb{R}^n$ has the standard Euclidean metric and $\Gamma$ is a lattice. The eigenvectors all have the form $e^{2 \pi i \langle v, w \rangle}$ where $v \in \mathbb{R}^n$ and $w \in \Gamma^{\vee}$ lies in the dual lattice. The corresponding eigenvalue is $-4 \pi^2 \| w \|^2$.
Among other things, this allows you to reduce the problem of finding two nonisometric isospectral manifolds to the problem of finding two nonisomorphic lattices with the same theta function.
Jeffrey Weeks has computed the spectra of homogeneous elliptic manifolds.
For arithmetic hyperbolic manifolds, the spectrum is in principle computable in the sense that one may define a Selberg zeta function arithmetically, whose roots give the spectrum.
Certain other homogeneous Heisenberg manifolds have their spectra computed.