There are a number of dimensions you can look at to give you a sense of what will be a reasonable algorithm to start with, namely:

• Number of training examples
• Dimensionality of the feature space
• Do I expect the problem to be linearly separable?
• Are features independent?
• Are features expected to linearly dependent with the target variable? *EDIT: see mycomment on what I mean by this
• Is overfitting expected to be a problem?
• What are the system's requirement in terms of speed/performance/memory usage...?
• ...

This list may seem a bit daunting because there are many issues that are not straightforward to answer. The good news though is, that as many problems in life, you can address this question by following the Occam's Razor principle: use the least complicated algorithm that can address your needs and only go for something more complicated if strictly necessary.
Logistic Regression
As a general rule of thumb, I would recommend to start with Logistic Regression. Logistic regression is a pretty well-behaved classification algorithm that can be trained as long as you expect your features to be roughly linear and the problem to be linearly separable. You can do some feature engineering to turn most non-linear features into linear pretty easily. It is also pretty robust to noise and you can avoid overfitting and even do feature selection by using l2 or l1 regularization. Logistic regression can also be used in Big Data scenarios since it is pretty efficient and can be distributed using, for example, ADMM (see logreg). A final advantage of LR is that the output can be interpreted as a probability. This is something that comes as a nice side effect since you can use it, for example, for ranking instead of classification.
Even in a case where you would not expect Logistic Regression to work 100%, do yourself a favor and run a simple l2-regularized LR to come up with a baseline before you go into using "fancier" approaches.
Ok, so now that you have set your baseline with Logistic Regression, what should be your next step. I would basically recommend two possible directions: (1) SVM's, or (2) Tree Ensembles. If I knew nothing about your problem, I would definitely go for (2), but I will start with describing why SVM's might be something worth considering.
Support Vector Machines
Support Vector Machines (SVMs) use a different loss function (Hinge) from LR. They are also interpreted differently (maximum-margin). However, in practice, an SVM with a linear kernel is not very different from a Logistic Regression (If you are curious, you can see how Andrew Ng derives SVMs from Logistic Regression in his Coursera Machine Learning Course). The main reason you would want to use an SVM instead of a Logistic Regression is because your problem might not be linearly separable. In that case, you will have to use an SVM with a non linear kernel (e.g. RBF). The truth is that a Logistic Regression can also be used with a different kernel, but at that point you might be better off going for SVMs for practical reasons. Another related reason to use SVMs is if you are in a highly dimensional space. For example, SVMs have been reported to work better for text classification.
Unfortunately, the major downside of SVMs is that they can be painfully inefficient to train. So, I would not recommend them for any problem where you have many training examples. I would actually go even further and say that I would not recommend SVMs for most "industry scale" applications. Anything beyond a toy/lab problem might be better approached with a different algorithm.
Tree Ensembles
This gets me to the third family of algorithms: Tree Ensembles. This basically covers two distinct algorithms: Random Forests and Gradient Boosted Trees. I will talk about the differences later, but for now let me treat them as one for the purpose of comparing them to Logistic Regression.
Tree Ensembles have different advantages over LR. One main advantage is that they do not expect linear features or even features that interact linearly. Something I did not mention in LR is that it can hardly handle categorical (binary) features. Tree Ensembles, because they are nothing more than a bunch of Decision Trees combined, can handle this very well. The other main advantage is that, because of how they are constructed (using bagging or boosting) these algorithms handle very well high dimensional spaces as well as large number of training examples.
As for the difference between Random Forests (RF) and Gradient Boosted Decision Trees (GBDT), I won't go into many details, but one easy way to understand it is that GBDTs will usually perform better, but they are harder to get right. More concretely, GBDTs have more hyper-parameters to tune and are also more prone to overfitting. RFs can almost work "out of the box" and that is one reason why they are very popular.
Deep Learning
Last but not least, this answer would not be complete without at least a minor reference toDeep Learning. I would definitely not recommend this approach as a general-purpose technique for classification. But, you might probably have heard how well these methods perform in some cases such as image classification. If you have gone through the previous steps and still feel you can squeeze something out of your problem, you might want to use a Deep Learning approach. The truth is that if you use an open source implementation such as Theano, you can get an idea of how some of these approaches perform in your dataset pretty quickly.
Summary
So, recapping, start with something simple like Logistic Regression to set a baseline and only make it more complicated if you need to. At that point, tree ensembles, and in particular Random Forests since they are easy to tune, might be the right way to go. If you feel there is still room for improvement, try GBDT or get even fancier and go for Deep Learning.
You can also take a look at the Kaggle Competitions. If you search for the keyword "classification" and select those that are completed, you will get a good sense of what people used to win competitions that might be similar to your problem at hand. At that point you will probably realize that using an ensemble is always likely to make things better. The only problem with ensembles, of course, is that they require to maintain all the independent methods working in parallel. That might be your final step to get as fancy as it gets.

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