There are several algorithms that students can and should be learning in an upperdivision mathematics curriculum. Most degrees seem to only touch a few algorithms outside of a numerical analysis course (e.g., Newton’s method, the Euclidian algorithm, and maybe the simplex method), but there are so many more that students need to see.
It’s a mistake to only put algorithms in elective numerical analysis courses. They should be integrated into the curriculum.
Most of the following are covered in BYU’s relatively new Applied and Computational Mathematics undergrad degree.
I would love to hear from people who want to add to the list, argue about the list, and/or have experienced with these and related topics.
 Encryption Algorithms: probabilistic primes (using Fermat’s little theorem), RSA, Euclidian algorithm, Diffie Hillman key exchange

Solving Linear Systems and Finding Eigenvalues: Row reduction, Jacobi, Gauss Seidel, Successive over relaxation (SOR), and Krylov methods such as Arnoldi, Lancos, GMRES.

Spectral Decomposition: FFT in 123 dimensions, convolution, windowing and other signal processing algorithms (just inner products and FFT)

State Estimation: Kalman, extended Kalman, particle filters, recursive least squares, etc.

Compression: Huffman, LWZ, wavelet approximation

Tree Search: AVL trees, BlackWhite trees, Btrees

Constrained Optimization: simplex, interior point, \ell^1
regularization (which includes compressed sensing) 
Sampling and Markov Chain Monte Carlo: Gibbs, Metropolis, MetropolisHastings

Matrix Decompositions: LU, QR, SVD, etc.

Graph Algorithms: Minimum Spanning tree, traveling salesman, breadthfirst search, depthfirst search

Unconstrained Optimization: Newton, BFGS, conjugategradient

Dynamic programming (backward iteration)

Classification: Logistic regression, random forests, support vector
machines, neural networks 
Multiarmed bandit problems and Markov Decision Processes
Others are:

ODE Solvers (RKF, DormandPrince)

PDE solvers: finite difference/element: Most of these are just linear algebra solvers

Pseudorandom number generation: Twistor

Time series: ARMA, ARIMA (these can be done with the Kalman filter, but almost nobody does it this way)

Splines/Interpolation: Most of these are linear algebra solvers, Chebyshev interpolation uses FFT, barycentric Lagrange interpolation