Josu has given an example of how using quantum gates you can get an entangled state. However, quantum gates are sort of "black-boxes".
The physical methods for creating entangled states and testing for entangled states are covered well on the relevant Wikipedia page:
Methods of creating entanglement:
Entanglement is usually created by direct interactions between
subatomic particles. These interactions can take numerous forms. One
of the most commonly used methods is spontaneous parametric
down-conversion to generate a pair of photons entangled in
polarisation. Other methods include the use of a fiber coupler to
confine and mix photons, photons emitted from decay cascade of the
bi-exciton in a quantum dot, the use of the Hong–Ou–Mandel effect,
etc., In the earliest tests of Bell's theorem, the entangled particles
were generated using atomic cascades.
It is also possible to create entanglement between quantum systems
that never directly interacted, through the use of entanglement
swapping. Two independently-prepared, identical particles may also be
entangled if their wave functions merely spatially overlap, at least
Testing a system for entanglement:
Systems which contain no entanglement are said to be separable. For $2$-Qubit and Qubit-Qutrit systems ($2 × 2$ and $2 × 3$ respectively) the simple Peres–Horodecki criterion provides both a necessary and a sufficient criterion for separability, and thus for detecting entanglement. However, for the general case, the criterion is merely a sufficient one for separability, as the problem becomes NP-hard. A numerical approach to the problem is suggested by Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement". Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached. An implementation of the algorithm (including a built-in Peres-Horodecki criterion testing) is brought in the "StateSeparator" web-app.
In 2016 China launched the world’s first quantum communications satellite. The $100m Quantum Experiments at Space Scale (QUESS) mission was launched on Aug 16, 2016, from the Jiuquan Satellite Launch Center in northern China at 01:40 local time.
For the next two years, the craft – nicknamed "Micius" after the ancient Chinese philosopher – will demonstrate the feasibility of quantum communication between Earth and space, and test quantum entanglement over unprecedented distances.
In the June 16, 2017, issue of Science, Yin et al. report setting a new quantum entanglement distance record of $1203$ km, demonstrating the survival of a $2$-photon pair and a violation of a Bell inequality, reaching a CHSH valuation of $2.37 ± 0.09$, under strict Einstein locality conditions, from the Micius satellite to bases in Lijian, Yunnan and Delingha, Quinhai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude.
You might also want to see:
How do I show that a two-qubit state is an entangled state?
How to show that an n-level system is entangled?