Preliminary facts
In functional sense, the sigmoid is a partial case of the softmax function, when the number of classes equals 2. Both of them do the same operation: transform the logits (see below) to probabilities.
In simple binary classification, there's no big difference between the two,
however in case of multinomial classification, sigmoid allows to deal
with non-exclusive labels (a.k.a. multi-labels), while softmax deals
with exclusive classes (see below).
A logit (also called a score) is a raw unscaled value associated with a class, before computing the probability. In terms of neural network architecture, this means that a logit is an output of a dense (fully-connected) layer.
Tensorflow naming is a bit strange: all of the functions below accept logits, not probabilities, and apply the transformation themselves (which is simply more efficient).
Sigmoid functions family
As stated earlier, sigmoid loss function is for binary classification.
But tensorflow functions are more general and allow to do
multi-label classification, when the classes are independent.
In other words, tf.nn.sigmoid_cross_entropy_with_logits solves N
binary classifications at once.
The labels must be one-hot encoded or can contain soft class probabilities.
tf.losses.sigmoid_cross_entropy in addition allows to set the in-batch weights,
i.e. make some examples more important than others.
tf.nn.weighted_cross_entropy_with_logits allows to set class weights
(remember, the classification is binary), i.e. make positive errors larger than
negative errors. This is useful when the training data is unbalanced.
Softmax functions family
These loss functions should be used for multinomial mutually exclusive classification,
i.e. pick one out of N classes. Also applicable when N = 2.
The labels must be one-hot encoded or can contain soft class probabilities:
a particular example can belong to class A with 50% probability and class B
with 50% probability. Note that strictly speaking it doesn't mean that
it belongs to both classes, but one can interpret the probabilities this way.
Just like in sigmoid family, tf.losses.softmax_cross_entropy allows
to set the in-batch weights, i.e. make some examples more important than others.
As far as I know, as of tensorflow 1.3, there's no built-in way to set class weights.
[UPD] In tensorflow 1.5, v2 version was introduced and the original softmax_cross_entropy_with_logits loss got deprecated. The only difference between them is that in a newer version, backpropagation happens into both logits and labels (here's a discussion why this may be useful).
Sparse functions family
Like ordinary softmax above, these loss functions should be used for
multinomial mutually exclusive classification, i.e. pick one out of N classes.
The difference is in labels encoding: the classes are specified as integers (class index),
not one-hot vectors. Obviously, this doesn't allow soft classes, but it
can save some memory when there are thousands or millions of classes.
However, note that logits argument must still contain logits per each class,
thus it consumes at least [batch_size, classes] memory.
Like above, tf.losses version has a weights argument which allows
to set the in-batch weights.
Sampled softmax functions family
These functions provide another alternative for dealing with huge number of classes.
Instead of computing and comparing an exact probability distribution, they compute
a loss estimate from a random sample.
The arguments weights and biases specify a separate fully-connected layer that
is used to compute the logits for a chosen sample.
Like above, labels are not one-hot encoded, but have the shape [batch_size, num_true].
Sampled functions are only suitable for training. In test time, it's recommended to
use a standard softmax loss (either sparse or one-hot) to get an actual distribution.
Another alternative loss is tf.nn.nce_loss, which performs noise-contrastive estimation (if you're interested, see this very detailed discussion). I've included this function to the softmax family, because NCE guarantees approximation to softmax in the limit.
tf.losses.log_loss, actually it's for binary crossentropy only. Also github.com/tensorflow/tensorflow/issues/2462 – Lexicology