Audio software companies have been striving to replicate the warmth and character of analog gear in a digital form for years. This has been a popular pursuit because while digital technology offers many benefits, it can sound sterile and lack the warmth and depth of analog equipment. To achieve this, software developers use a variety of techniques, such as circuit-modelling and black-box trial-and-error. However you can emulate some of these approaches using the stock plugins in your DAW.
Saturation
Saturation is a technique that has been used to give recordings warmth and character since the early days of analog recording. Essentially, it involves adding additional harmonics to a signal, which makes it sound fuller and more complex. Physically modelling a circuit requires a lot of attention to detail, but you can get 80% of the way there using a simple Hyperbolic Tangent (tanh) function. This is essentially a soft-clipper that generates additional harmonics when driven. The reason tanh is used, is because it generates even harmonics, which are very pleasing to the human ear. Note: Heavily saturating a signal can introduce unwanted low-frequencies via Aliasing, ensure you're using a saturator that has an Oversampling mode.
Frequency Response
Old analog equipment often lacks the digital clarity in the higher end of the frequency curve. This soft rolloff, when coupled with a powerful low-end, is what creates that "warm" and "full" sound (developers love using buzzwords to help market their products). To emulate this, we can rolloff higher frequencies using a high-shelf, and boost lower ones with a low-shelf. It's easy, and helps to create a more natural, balanced sound that is similar to what you might hear from analog gear.
Unstable, Changing Electronics
Electrical circuits can change over time, as tubes warm up and cool down, the signal going through them can change accordingly. Inside some of the more advanced whitebox models are interdependant parameters and non-linearities that change the sound depending on certain circumstances. While we can't necessarily emulate this inside a DAW, we can introduce some movement in the sound via Automation. Adjusting certain plugin parameters in a verse or chorus, such as adding slightly more high-end or decreasing saturation can add some interest in the track. Automation in general is an extremely powerful tool that many producers overlook, so get automating!In conclusion, the pursuit of the analog sound has been a driving force in the development of audio software for many years. While it may be impossible to completely replicate the sound of analog gear, these techniques have helped to create digital plugins that come very close.
One thing I've been following very closely (VERY CLOSELY) is the application of Neural Synthesis for tasks like analog modelling. Developers like Christian Steinmetz and Jatin Chowdhury have already created models using Differentiable Digital Signal Processing (DDSP) as well as good old-fashioned ConvNets and RNN's to create all sorts of interesting (and realistic) models. As these tools become more fleshed-out, I wouldn't be surprised if we start seeing them more often.